Effective medium theory of checkboard structures in the long-wavelength limit

Zhanlei Hao; Yawen Zhuang; Ying Chen; Yineng Liu; Huanyang Chen

doi:10.3788/COL202018.072401

Journals >Chinese Optics Letters >Volume 18 >Issue 7 >Page 072401 > Article

Chinese Optics Letters
Vol. 18, Issue 7, 072401 (2020)

Effective medium theory of checkboard structures in the long-wavelength limit

Zhanlei Hao, Yawen Zhuang, Ying Chen^*, Yineng Liu^**, and Huanyang Chen^***

Author Affiliations

Institute of Electromagnetics and Acoustics and Key Laboratory of Electromagnetic Wave Science and Detection Technology, Xiamen University, Xiamen 361005, China

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DOI: 10.3788/COL202018.072401 Cite this Article Set citation alerts

Zhanlei Hao, Yawen Zhuang, Ying Chen, Yineng Liu, Huanyang Chen. Effective medium theory of checkboard structures in the long-wavelength limit[J]. Chinese Optics Letters, 2020, 18(7): 072401 Copy Citation Text

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Abstract

Effective medium theory is a powerful tool to solve various problems for achieving multifarious functionalities and applications. In this article, we present a concise empirical formula about effective permittivity of checkboard structures for different directions. To verify our empirical formula, we perform simulations of checkboard periodic structures in squares, rectangles, and sectors in two dimensions. Our results show that the formula is valid in a large range of parameters. This work provides a new way to understand and design composite materials, which might lead to further optical applications in transformation optics.

Keywords

checkboard structures effective medium theory effective permittivity

\nabla \cdot D^{*} = \sqrt{ϵ_{1} \cdot ϵ_{2}} \nabla \cdot (e_{z} \times E) = - \sqrt{ϵ_{1} \cdot ϵ_{2}} e_{z} \cdot \nabla \times E = 0, \nabla \times E^{*} = \frac{1}{\sqrt{ϵ_{1} \cdot ϵ_{2}}} \nabla \times (e_{z} \times D) = \frac{1}{\sqrt{ϵ_{1} \cdot ϵ_{2}}} [e_{z} \nabla \cdot D - D \nabla \cdot e_{z} + (D \cdot \nabla) e_{z} - (e_{z} \cdot \nabla) D] = 0, D^{*} = \sqrt{ϵ_{1} \cdot ϵ_{2}} e_{z} \times E = \frac{\sqrt{ϵ_{1} \cdot ϵ_{2}}}{ϵ} e_{z} \times D = \frac{ϵ_{1} \cdot ϵ_{2}}{ϵ} E^{*},

(1)

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\bar{ϵ} = ϵ_{eff} = \sqrt{ϵ_{1} \cdot ϵ_{2}} .

(2)

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ϵ_{eff} = [\begin{matrix} ϵ_{x} & 0 \\ 0 & ϵ_{y} \end{matrix}] = [\begin{matrix} \sqrt{ϵ_{1} ϵ_{2} e^{0.28 \ln \frac{ϵ_{1}}{ϵ_{2}} \ln \frac{a}{b}}} & 0 \\ 0 & \sqrt{\frac{ϵ_{1} ϵ_{2}}{e^{0.28 \ln \frac{ϵ_{1}}{ϵ_{2}} \ln \frac{a}{b}}}} \end{matrix}] .

(3)

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Zhanlei Hao, Yawen Zhuang, Ying Chen, Yineng Liu, Huanyang Chen. Effective medium theory of checkboard structures in the long-wavelength limit[J]. Chinese Optics Letters, 2020, 18(7): 072401

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