# Effective Permittivity for FDTD Calculation of Plasmonic Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Effective Permittivity for Plasmonic Materials

## 3. Numerical Validations

#### 3.1. Infinitely-Long Ag Cylinder

#### 3.2. Au Sphere

## 4. Conclusions

## Acknowledgments

## References

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**Figure 1.**Partially filled FDTD cells in the TM mode: (

**a**) interface parallel to ${E}_{x}$; (

**b**) interface perpendicular to ${E}_{x}$. Dotted lines represent the integration lines in Ampere’s and Faraday’s laws. $d,f$ represent filling factors on the integration lines. ${\epsilon}_{1},{\epsilon}_{2}$ are permittivities outside and inside the plasmonic material, respectively.

**Figure 2.**Surface plasmon (SP) resonances: (

**a**) propagating SP; (

**b**) localized SP. Electric charge distributions and lines of force are depicted. Regions enclosed with dotted lines indicate strongly resonant SP fields.

**Figure 3.**Simulation setup for an infinitely-long Ag cylinder. Incident TM polarization ($a=$ radius, $h=$ grid spacing). Ag cylinder (gray region) is immersed in air (white region). FDTD results are compared with Mie theory on dotted contour C.

**Figure 4.**Calculated E

_{y}scattered intensity distributions for an infinitely-long Ag cylinder using: (a) Mie theory; (b) Staircase model; (c) EP model; (d) S-EP model (black region represents the cylinder); (e) Angular intensity distributions on contour C of radius 1.2a.

**Figure 5.**Root-mean-square (RMS) errors of staircase, EP, and S-EP models relative to Mie theory on contour C as a function of: (a) incident wavelength (h = 20 nm, a = 538.126 nm); (b) grid spacing (λ

_{0}= 430.501 nm, a = 538.126 nm); (c) cylinder radius (h = 20 nm, λ

_{0}= 430.501 nm); (d) subwavelength cylinder radius (h = 10 nm, λ

_{0}= 430.501 nm).

**Figure 6.**Simulation setup for a Au sphere ($a=$ radius, $h=$ grid spacing). Incident ${E}_{y}$ polarized wave propagates along the z axis. Au sphere (gray region) is immersed in air (white region). FDTD results are compared with Mie theory on surface S of radius $1.2a$ and contour C on S in the y-z plane (dotted line).

**Figure 7.**Calculated ${E}_{y}$ scattered intensity distributions for a Au sphere on surface S of radius $1.2a$ using: (

**a**) Mie theory; (

**b**) Staircase model; (

**c**) EP model; (

**d**) S-EP model; (

**e**) Angular intensity distributions on contour C in the y-z plane.

Parameter | Input Value |
---|---|

permittivity of air ${\epsilon}_{1}$ | 1 |

permittivity of Ag ${\epsilon}_{2}$ | −6.06 + i0.197 |

wavelength ${\lambda}_{0}$ | 430.501 nm |

cylinder radius a | 538.126 nm |

grid spacing h | 20 nm |

Parameter | Input Value |
---|---|

permittivity of air ${\epsilon}_{1}$ | 1 |

permittivity of Au ${\epsilon}_{2}$ | −10.662 + i1.374 |

wavelength ${\lambda}_{0}$ | 616.837 nm |

cylinder radius a | 925.255 nm |

grid spacing h | 20 nm |

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Okada, N.; Cole, J.B.
Effective Permittivity for FDTD Calculation of Plasmonic Materials. *Micromachines* **2012**, *3*, 168-179.
https://doi.org/10.3390/mi3010168

**AMA Style**

Okada N, Cole JB.
Effective Permittivity for FDTD Calculation of Plasmonic Materials. *Micromachines*. 2012; 3(1):168-179.
https://doi.org/10.3390/mi3010168

**Chicago/Turabian Style**

Okada, Naoki, and James B. Cole.
2012. "Effective Permittivity for FDTD Calculation of Plasmonic Materials" *Micromachines* 3, no. 1: 168-179.
https://doi.org/10.3390/mi3010168