## Novel approach to photonic bands with frequency-dependent dielectric constants

Optics Express, Vol. 3, Issue 1, pp. 12-18 (1998)

http://dx.doi.org/10.1364/OE.3.000012

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### Abstract

We formulated a novel method to calculate the dispersion relations of arbitrary photonic crystals with frequency-dependent dielectric constants based on the numerical simulation of dipole radiation. As an example, we applied this method to a two-dimensional square lattice of metallic cylinders and obtained a good agreement with the previous result by means of the plane-wave expansion method by Kuzmiak *et al*. [Phys. Rev. B **50**, 16 835 (1994)]. In addition to the dispersion relations, we could obtain the symmetries and the wave functions of the eigenmodes.

© Optical Society of America

## 1. Introduction

4. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. **67**, 2017–2020 (1991). [CrossRef] [PubMed]

5. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. **67**, 3380–3383 (1991). [CrossRef] [PubMed]

6. E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B **10**, 283–295 (1993). [CrossRef]

*et al*.[7

7. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B **50**, 16 835–16 844 (1994). [CrossRef]

*E*polarization) and the plane-wave expansion method gives a good convergence. But we have to devise another method when the wave equation does not lead to a well-defined eigenvalue problem, and then the good convergence of the plane-wave expansion is not necessarily guaranteed.

8. K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B **56**, 4830–4835 (1997). [CrossRef]

9. K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B **56**, 14 905–14 908 (1997). [CrossRef]

*et al*, [7

7. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B **50**, 16 835–16 844 (1994). [CrossRef]

*E*polarization. The numerical results will be presented and compared with those of Kuzmiak

*et al*in Sec. 3.

## 2. Theory

**P**

_{d}(

**r**,

*t*) that was embedded in the photonic crystal with a structural defect:

**, and**

*μ***r**

_{0}are the magnitude and the position of the dipole moment, respectively,

*ω*is the angular frequency of the oscillation, and

*δ*is Dirac’s delta function. Then we showed by means of a Green’s-function method[10

10. K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B **54**, 5732–5741 (1996). [CrossRef]

**E**(

**r**,

*t*), and the electromagnetic energy radiated in a unit time,

*U*, are given as follows.[8

8. K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B **56**, 4830–4835 (1997). [CrossRef]

*ω*and

_{d}**E**

_{d}(

**r**) denote the eigen-angular frequency and the eigenfunction of the relevant defect mode,

*γ*is its decay rate,

*V*is the volume of the photonic crystal.

**E**

_{d}(

**r**) is normalized as follows.

**r**) is the position-dependent dielectric constant of the photonic crystal. When we derived Eqs. (2) and (3), we assumed that

*ω*was close to

*ω*that was isolated in the photonic band gap and neglected the contribution from all other eigenmodes. If we can evaluate the frequency dependence of

_{d}*U*, we can obtain the eigen-angular frequency as the resonance frequency [see Eq. (3)], and then the induced electric field is proportional to the eigenfunction [see Eq. (2)]. Then we discretized the wave equation derived from the Maxwell’s equations to obtain a difference equation and solved it numerically. We could obtain

*ω*as the resonance frequency as we had expected, and the calculated values were very close to observed ones. [8

_{d}8. K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B **56**, 4830–4835 (1997). [CrossRef]

9. K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B **56**, 14 905–14 908 (1997). [CrossRef]

*c*is the light velocity in vacuum. In Eq. (6),

**P**

_{d}(

**r**,

*t*) stands for the virtual oscillating dipole moment as before and

**D**

_{0}(

**r**,

*t*) denotes the electric displacement due to the regular dielectric structure of the photonic crystal. The latter is generally given by the convolution integral of the electric field

**E**(

**r**,

*t*) and the dielectric response function Φ(

**r**,

*t*):

**r**,

*t*) is given by the Fourier transform of the dielectric constant ϵ(

**r**,

*ω*), which is now a function of the frequency as well as the spatial coordinates:

*et al*.[7

7. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B **50**, 16 835–16 844 (1994). [CrossRef]

*ω*is the plasma frequency,

_{p}*τ*is the relaxation time, and

*ϵ*

_{∞}is the dielectric constant at sufficiently high frequencies. In Eq. (9), we took into account the imaginary part of the dielectric constant in order to fulfill the Kramers-Kronig relation, and hence, the causality. Then Eq. (8) leads to

*θ*(

*t*) is a step function. From Eqs. (5), (6), (7), and (10), we obtain the following wave equation for the metallic region.

*ϵ*(

**r**,

*ω*) = 1 (air) outside the metal. Then for this region,

**E**(

**r**, 0) = 0 and a boundary condition

**k**is a wave vector in the first Brillouin zone and

**a**is the elementary lattice vector. The latter condition extracts the contribution to the radiated electromagnetic field from particular eigenmodes with the specified wave vector. Therefore, we can calculate the resonance frequency as a function of

**k**, i.e., we can obtain the dispersion relations. According to Kuzmiak

*et al*.,[7

**50**, 16 835–16 844 (1994). [CrossRef]

*R*for the numerical calculation. The following parameters were assumed:

*R*/

*a*= 0.472,

*ϵ*

_{∞}= 1.0, and

*ω*

_{p}*a*/27

*πc*= 1.0, where

*a*denotes the lattice constant. Because of the boundary condition, Eq. (13), it was enough to deal with only one unit cell, and therefore, the CPU time necessary for the numerical calculation was small. In the actual calculation, the unit cell was divided into 40 × 40 parts, and one period of the oscillation was divided into 160 parts in order to discretize the wave equation. The further decrease of the size of the spatial and temporal meshes did not bring about an apparent change in the resonance frequencies.

## 3. Results and discussion

**k**= 0. The abscissa represents the normalized frequency. □, ○, ◇, and • denote the accumulated electromagnetic energy after 10, 20, 50, and 100 cycles of the oscillation, respectively. A resonance at

*ωa*/2

*πc*= 0.745 is clearly observed in this figure.

*ωa*/2

*πc*< 0.745. We can show that this cut-off frequency is consistent with the result of the long-wavelength approximation of the Maxwell’s equations. When we compare this figure with Fig. 1(b) of Ref. [7

**50**, 16 835–16 844 (1994). [CrossRef]

*E*representation of the

*C*

_{4v}point group that is doubly degenerate. Therefore, we believe that our calculation is more accurate concerned with this matter.

*E*representation. Finally, Fig. 4 shows the field distribution of the eigenmodes at the

*X*point. They are consistent with the symmetry assignments given in Fig. 2.

*γ*of the eigenmodes in this paper because we were mainly interested in the evaluation of the accuracy and efficiency of the present method as a tool for the photonic band calculation.

*γ*is inevitably non-zero when the dielectric constant depends on frequency and has the non-zero imaginary part as a result of the Kramers-Kronig relation.

*γ*can be evaluated by analysing the width of the resonance curve, or more practically by calculating the temporal evolution of the accumulated electromagnetic energy after switching off the oscillation of the dipole.

## 4. Conclusion

## Acknowledgments

## References

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

2. | Photonic Band Gaps and Localization, edited by C. M. Soukoulis (Plenum, New York, 1993). |

3. | Photonic Band Gap Materials, edited by C. M. Soukoulis (Kluwer, Dordrecht, 1996). |

4. | S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. |

5. | E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. |

6. | E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Am. B |

7. | V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B |

8. | K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B |

9. | K. Sakoda, T. Ueta, and K. Ohtaka, “Numerical analysis of eigenmodes localized at line defects in photonic lattices,” Phys. Rev. B |

10. | K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Focus Issue: Photonic crystals

**History**

Original Manuscript: March 23, 1998

Revised Manuscript: February 16, 1998

Published: July 6, 1998

**Citation**

Kazuaki Sakoda and Jun Kawamata, "Novel approach to photonic bands with
frequency-dependent dielectric constants," Opt. Express **3**, 12-18 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-1-12

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, (Princeton University Press, Princeton, 1995).
- Photonic Band Gaps and Localization, edited by C. M. Soukoulis (Plenum, New York, 1993).
- Photonic Band Gap Materials, edited by C. M. Soukoulis (Kluwer, Dordrecht, 1996).
- S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, "Microwave propagation in two-dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991). [CrossRef] [PubMed]
- E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, "Donor and acceptor modes in photonic band structure," Phys. Rev. Lett. 67, 3380-3383 (1991). [CrossRef] [PubMed]
- E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993). [CrossRef]
- V. Kuzmiak, A. A. Maradudin, and F. Pincemin, "Photonic band structures of two-dimensional systems containing metallic components," Phys. Rev. B 50, 16 835-16 844 (1994). [CrossRef]
- K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997). [CrossRef]
- K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14 905-14 908 (1997). [CrossRef]
- K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell's equations and their applications," Phys. Rev. B 54, 5732-5741 (1996). [CrossRef]

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