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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(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 ϵ1 and ϵ2, respectively.

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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The field patterns Hz when incident with (a) a plane wave and (c) a point source on a square CS, with 50×50 unit cells in the middle (only the sketch is shown here). The period a is set as 200 nm, with ϵ1=4 and ϵ2=1. (b), (d) Field patterns when using the isotropic and uniform medium in the middle region.

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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$ϵ_{1} / ϵ_{2}$	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 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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$ϵ$	a/b
$ϵ$	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

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

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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