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

Chinese Optics Letters, Volume. 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^***

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

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Figures & Tables(6)

Fig. 1. (a) Square CS with a period $a$ . (b) Rectangular CS with a period $a$ along the $x$ direction and $a$ period $b$ along the $y$ direction. The permittivity of the yellow region and white region is $ϵ_{1}$ and $ϵ_{2}$ , respectively.

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Fig. 2. The field patterns $H_{z}$ when incident with (a) a plane wave and (c) a point source on a square CS, with $50 \times 50$ unit cells in the middle (only the sketch is shown here). The period $a$ is set as 200 nm, with $ϵ_{1} = 4$ and $ϵ_{2} = 1$ . (b), (d) Field patterns when using the isotropic and uniform medium in the middle region.

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Fig. 3. The field patterns $H_{z}$ when incident with (a) a plane wave and (c) a point source to the middle bulk material composed with rectangular CS. The middle region of the effective medium has $50 \times 100$ unit cells (only the sketch is shown here), with $a = 200 nm$ , $b = 100 nm$ , $ϵ_{1} = 4$ , and $ϵ_{2} = 1$ . (b), (d) Field patterns when using the anisotropic and uniform medium.

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Fig. 4. Field patterns $H_{z}$ when (a) a plane wave and (c) a cylindrical wave pass through a gradient circular ring. The circular grids (only the sketch is shown here) are formed by many sector units in the CS, including $10 (r) \times 24 (θ)$ unit cells. Inner and outer radii of this ring are 1 μm and 4 μm, and we keep the permittivity of $ϵ_{1} = 4$ and $ϵ_{2} = 1$ unchanged in each unit cell. (b), (d) Field patterns when using a gradient and anisotropic medium incident by (b) a plane wave or (d) a point source, where the same layers along the radial direction are set.

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Table 1. Effective Permittivity $ϵ_{x}$ and $ϵ_{y}$ Obtained from the Slopes along $Γ X$ and $Γ Y$ Directions in the Band Structure, While Keeping $ϵ_{1} / ϵ_{2}$ Unchanged for Each Row and Tuning the Ratios of $a / b$ from 0.5 to 4

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Table 1. Effective Permittivity $ϵ_{x}$ and $ϵ_{y}$ Obtained from the Slopes along $Γ X$ and $Γ Y$ Directions in the Band Structure, While Keeping $ϵ_{1} / ϵ_{2}$ Unchanged for Each Row and Tuning the Ratios of $a / b$ from 0.5 to 4

$ϵ_{1} / ϵ_{2}$	a/b
$ϵ_{1} / ϵ_{2}$	4	2	1	0.8	0.65	0.5
5	$ϵ_{x} = 3.170$	$ϵ_{x} = 2.604$	$ϵ_{x} = 2.236$	$ϵ_{x} = 2.173$	$ϵ_{x} = 2.101$	$ϵ_{x} = 1.946$
5	$ϵ_{y} = 1.810$	$ϵ_{y} = 1.946$	$ϵ_{y} = 2.236$	$ϵ_{y} = 2.368$	$ϵ_{y} = 2.463$	$ϵ_{y} = 2.605$
4.5	$ϵ_{x} = 2.837$	$ϵ_{x} = 2.411$	$ϵ_{x} = 2.117$	$ϵ_{x} = 2.021$	$ϵ_{x} = 1.949$	$ϵ_{x} = 1.838$
4.5	$ϵ_{y} = 1.780$	$ϵ_{y} = 1.839$	$ϵ_{y} = 2$ .117	$ϵ_{y} = 2.219$	$ϵ_{y} = 2.307$	$ϵ_{y} = 2.411$
4	$ϵ_{x} = 2.650$	$ϵ_{x} = 2.218$	$ϵ_{x} = 2.017$	$ϵ_{x} = 1.919$	$ϵ_{x} = 1.891$	$ϵ_{x} = 1.733$
4	$ϵ_{y} = 1.719$	$ϵ_{y} = 1.733$	$ϵ_{y} = 2.017$	$ϵ_{y} = 2.087$	$ϵ_{y} = 2.155$	$ϵ_{y} = 2.218$
3	$ϵ_{x} = 2.131$	$ϵ_{x} = 1.832$	$ϵ_{x} = 1.696$	$ϵ_{x} = 1.650$	$ϵ_{x} = 1.605$	$ϵ_{x} = 1.522$
3	$ϵ_{y} = 1.568$	$ϵ_{y} = 1.521$	$ϵ_{y} = 1.696$	$ϵ_{y} = 1.767$	$ϵ_{y} = 1.818$	$ϵ_{y} = 1.832$
2.5	$ϵ_{x} = 1.871$	$ϵ_{x} = 1.639$	$ϵ_{x} = 1.564$	$ϵ_{x} = 1.517$	$ϵ_{x} = 1.483$	$ϵ_{x} = 1.415$
2.5	$ϵ_{y} = 1.463$	$ϵ_{y} = 1.415$	$ϵ_{y} = 1.564$	$ϵ_{y} = 1.607$	$ϵ_{y} = 1.645$	$ϵ_{y} = 1.639$
2	$ϵ_{x} = 1.611$	$ϵ_{x} = 1.446$	$ϵ_{x} = 1.414$	$ϵ_{x} = 1.395$	$ϵ_{x} = 1.392$	$ϵ_{x} = 1.308$
2	$ϵ_{y} = 1.357$	$ϵ_{y} = 1.308$	$ϵ_{y} = 1.414$	$ϵ_{y} = 1.447$	$ϵ_{y} = 1.453$	$ϵ_{y} = 1.446$
1.5	$ϵ_{x} = 2.654$	$ϵ_{x} = 2.521$	$ϵ_{x} = 2.451$	$ϵ_{x} = 2.427$	$ϵ_{x} = 2.396$	$ϵ_{x} = 2.353$
1.5	$ϵ_{y} = 2.275$	$ϵ_{y} = 2.353$	$ϵ_{y} = 2.451$	$ϵ_{y} = 2.457$	$ϵ_{y} = 2.496$	$ϵ_{y} = 2.525$

Table 2. Fitted Anisotropic Permittivity $ϵ_{x}$ and $ϵ_{y}$ of the Circular Ring Material Based on Eq. (3), Where the Permittivity $ϵ_{1} = 4$ and $ϵ_{2} = 1$ Are Kept Unchanged in Each Unit Cell
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Table 2. Fitted Anisotropic Permittivity $ϵ_{x}$ and $ϵ_{y}$ of the Circular Ring Material Based on Eq. (3), Where the Permittivity $ϵ_{1} = 4$ and $ϵ_{2} = 1$ Are Kept Unchanged in Each Unit Cell
$ϵ$ a/b
1 1.25 1.54 1.80 2.10 2.33 2.57 2.88 3.10 3.33
$ϵ_{x}$ 2.001 2.093 2.171 2.239 2.299 2.353 2.403 2.448 2.491 2.530
$ϵ_{y}$ 1.999 1.911 1.842 1.787 1.740 1.700 1.665 1.634 1.606 1.581

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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," Chin. Opt. Lett. 18, 072401 (2020)

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Paper Information

Category: Plasmonics and Metamaterials

Received: Mar. 16, 2020

Accepted: Apr. 30, 2020

Posted: May. 6, 2020

Published Online: May. 27, 2020

The Author Email: Ying Chen (lyn610@xmu.edu.cn), Yineng Liu (irusying1218@xmu.edu.cn), Huanyang Chen (kenyon@xmu.edu.cn)

DOI:10.3788/COL202018.072401

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology

Table 1. Effective Permittivity ϵx and ϵy Obtained from the Slopes along ΓX and ΓY Directions in the Band Structure, While Keeping ϵ1/ϵ2 Unchanged for Each Row and Tuning the Ratios of a/b from 0.5 to 4

Table 1. Effective Permittivity ϵx and ϵy Obtained from the Slopes along ΓX and ΓY Directions in the Band Structure, While Keeping ϵ1/ϵ2 Unchanged for Each Row and Tuning the Ratios of a/b from 0.5 to 4

Table 2. Fitted Anisotropic Permittivity ϵx and ϵy of the Circular Ring Material Based on Eq. (3), Where the Permittivity ϵ1=4 and ϵ2=1 Are Kept Unchanged in Each Unit Cell

Table 2. Fitted Anisotropic Permittivity ϵx and ϵy of the Circular Ring Material Based on Eq. (3), Where the Permittivity ϵ1=4 and ϵ2=1 Are Kept Unchanged in Each Unit Cell

Table 1. Effective Permittivity $ϵ_{x}$ and $ϵ_{y}$ Obtained from the Slopes along $Γ X$ and $Γ Y$ Directions in the Band Structure, While Keeping $ϵ_{1} / ϵ_{2}$ Unchanged for Each Row and Tuning the Ratios of $a / b$ from 0.5 to 4

Table 1. Effective Permittivity $ϵ_{x}$ and $ϵ_{y}$ Obtained from the Slopes along $Γ X$ and $Γ Y$ Directions in the Band Structure, While Keeping $ϵ_{1} / ϵ_{2}$ Unchanged for Each Row and Tuning the Ratios of $a / b$ from 0.5 to 4

Table 2. Fitted Anisotropic Permittivity $ϵ_{x}$ and $ϵ_{y}$ of the Circular Ring Material Based on Eq. (3), Where the Permittivity $ϵ_{1} = 4$ and $ϵ_{2} = 1$ Are Kept Unchanged in Each Unit Cell

Table 2. Fitted Anisotropic Permittivity $ϵ_{x}$ and $ϵ_{y}$ of the Circular Ring Material Based on Eq. (3), Where the Permittivity $ϵ_{1} = 4$ and $ϵ_{2} = 1$ Are Kept Unchanged in Each Unit Cell