2018 Thailand-Japan MicroWave (TJMW2018)

2-D Flat Lenses Based on Transformation Electromagnetics Tsutomu NAGAYAMA Seiji FUKUSHIMA and Toshio WATANABE

Graduate School of Science and Engineering, Kagoshima University 1-21-40 Korimoto, Kagoshima-shi, Kagoshima, 890-0065 Japan

E-mail: {t-nagayama, fukushima, wata104}@eee.kagoshima-u.ac.jp

Abstract 2-D flat lenses based on transformation electromagnetics are presented. These are lenses mimicking original lenses and are independent of their physical shapes. In this paper, two lenses mimicking a convex lens and a concave lens are designed with a distributed anisotropic circuit model based on the transmission-line approach. These operations are confirmed by circuit simulations.

Keyword Flat lens, Transformation electromagnetics, Metamaterials

1. INTRODUCTION By using a medium composed of metamaterials

with highly controlled constitutive parameters corresponding to an arbitrary coordinate system, we can highly control electromagnetic waves. The concept is referred to as transformation electromagnetics [1]–[21]. As applications of this, a cloak of invisibility hiding an object [1]–[18] and an illusion medium mimicking an arbitrary object [19] have been known. Also, by using the technique based on transformation electromagnetics, there is a possibility that electromagnetic devices, which are independent of their physical shapes, can be realized [20], [21].

In this paper, as novel applications, two 2-D flat lenses, which operates as if these are a convex lens and a concave lens, are presented based on transformation electromagnetics. By using a design method based on the transmission-line approach proposed by author [17], [18], these lenses are designed. In Section II, those concepts are presented. In Section III, those lenses are designed with a distributed anisotropic circuit model [18]. In section IV, those operations are confirmed by circuit simulations.

2. CONCEPTS OF 2-D FLAT LENSES

Figure 1(a) and (b) shows a 2-D original convex lens and a mimicking flat lens, respectively. Also, Fig. 2(a) and (b) shows a 2-D original concave lens and a mimicking flat lens, respectively. Based on the concept of transformation electromagnetics, two flat lenses can be realized by transforming the rectangular space including those original lenses into another rectangular space.

First, we consider the coordinate transformation from the original Cartesian coordinate system (x, y) including an original convex lens shown in Fig. 3(a) into the conformal coordinate system (x ′, y′) shown in Fig. 3(b) with relations:

, , (1) ʹx =1αx ʹy = y

(a) (b)

Fig. 1. Concept of a flat lens mimicking a convex lens. (a) Original convex lens to be mimicked. (b) Flat lens mimicking the convex lens.

(a) (b)

Fig. 2. Concept of a flat lens mimicking a concave lens. (a) Original concave lens to be mimicked. (b) Flat lens mimicking the concave lens.

2018 Thailand-Japan MicroWave (TJMW2018)

where –Tc £ x £ 0, –Dconvex/2 £ y £ Dconv ex/2, –Tc/a £ x' £ 0, and –Dconv ex/2 £ y' £ Dconvex/2. Next, we also consider the coordinate transformation from the original Cartesian coordinate system (x, y) including an original concave lens shown in Fig. 4(a) into the conformal coordinate system (x′, y ′) shown in Fig. 4(b) with the same relations as (1) where –Te £ x £ 0, –Dcon cav e /2 £ y £ Dcon cav e /2, –Te/a £ x' £ 0, and –Dcon cav e /2 £ y' £ Dcon cave /2. In the following, we choose the refractive indices of both original lenses as n = 1.5 and design two flat lenses with Tc = 0.289Dconv ex, Te = 0.287Dcon cav e, and a = 2.0. 3. DESIGN WITH A DISTRIBUTED ANISOTROPIC

CIRCUIT MODEL According to [1]–[3], transformations for two flat

lenses can be realized by media having the material tensor parameters:

, , (2)

, , (3)

where z-polarized TE incidence is assumed. Equation (2) corresponds to material tensor parameters in the area where the original lens is not included in the area before the transformation. Also, (3) corresponds to material tensor parameters in the other areas. Here, we design two flat lenses with a distributed anisotropic circuit model shown in Fig. 5. The

characteristic impedances and the electrical lengths equivalent to material tensor parameters of (2) and (3) can be calculated from the following design formulas [18]:

, (4)

, (5)

, (6)

where Dd is the unit cell length and the electrical lengths are defined as bxlx = byly ≡ bl.

µ =µxx µxy

µ yx µ yy

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

1α

0

0 α

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ε = ε z =α

µ =µxx µxy

µ yx µ yy

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

1α

0

0 α

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ε = ε z = n2α

µxx =2Z0 yΔd

tanβl

µ yy =2Z0xΔd

tanβl

ε z =Y0x +Y0 yΔd

sinβl

(a) (b) Fig. 3. Coordinate transformation for a flat lens mimicking a convex lens. (a) Original Cartesian coordinate system (x, y) including a convex lens. (b) Transformed coordinate system (x′, y ′) for a mimicking flat lens.

(a)

-Tc 0x

Dconvex/2

- Dconvex/2

y 0

-Tc/a 0xʹ

yʹ

Dconvex/2

- Dconvex/2

0

(a) (b) Fig. 4. Coordinate transformation for a flat lens mimicking a concave lens. (a) Original Cartesian coordinate system (x, y) including a concave lens. (b) Transformed coordinate system (x ′, y′) for a mimicking flat lens.

(a)

-Te 0x

Dconcave/2

- Dconcave/2

y 0

-Te/a 0xʹ

yʹ

Dconcave/2

- Dconcave/2

0

Fig. 5. Distributed anisotropic circuit model [18].

Z 0x, b xl x/2

Z 0x, b xl x/2

Z0y , b

y ly /2

Z0y , b

y ly /2

xy

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4. CIRCUIT SIMULATIONS First, we carry out circuit simulations for the flat

lens mimicking the convex lens. In this simulation, Tc is chosen as 50Dd. Calculated voltage amplitude and phase distributions of the center nodes in the unit cells for the cases with the original convex lens and the mimicking flat lens are shown in Fig. 6(a) and (b), respectively. The wavelength is chosen as l = 24Dd. By comparing Fig. 6(a) with (b), it is seen that the flat lens mimics the operation of the original convex lens well.

Next, circuit simulations for the flat lens mimicking the concave lens are also carried out. In this simulation, Te is chosen as 50Dd. Fig. 7(a) and (b) shows calculated voltage amplitude and phase distributions for the case with the original concave lens and the mimicking flat lens, respectively. The wavelength is also chosen as l = 24Dd. These results also show that the operation of the original concave lens can be mimicked well by the flat lens.

5. CONCLUSIONS

2-D flat lenses mimicking a convex lens and a concave lens have been presented based on transformation electromagnetics. These lenses have been designed with a distributed anisotropic circuit model based on the transmission-line approach. These operations have been confirmed by circuit simulations. References [1] J. B. Pendry, D. Schurig, and D. R. Smith,

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(a)

(b)

Fig. 6. Complex voltage distributions (l = 24Dd). Left: Amplitude. Right: Phase. (a) Original convex lens. (b) Mimicking flat lens.

Min Max -p p

Min Max -p p

(a)

(b)

Fig. 7. Complex voltage distributions (l = 24Dd). Left: Amplitude. Right: Phase. (a) Original concave lens. (b) Mimicking flat lens.

Min Max -p p

Min Max -p p

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