Asymmetric transmission with double bands based on chiral twisted double-split-ring resonators

Optics Communications 323 (2014) 19–22

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Asymmetric transmission with double bands based on chiral twisted double-split-ring resonators Dao-Ya Liu, Xiao-Min Zhai, Li-Fang Yao, Jian-Feng Dong n Institute of Optical Fiber Communication and Network Technology, Ningbo University, Ningbo 315211, China

art ic l e i nf o

a b s t r a c t

Article history: Received 22 October 2013 Received in revised form 20 February 2014 Accepted 22 February 2014 Available online 6 March 2014

A bi-layered chiral structure with double split-ring resonators is proposed to achieve dual-band asymmetric transmission of linear polarization waves propagating forward and backward. Moreover, according to the theoretical and numerical results, this designed chiral metamaterial can achieve the asymmetric transmission of both linearly and circularly polarized waves, in contrast to previously reported chiral structures that can realize asymmetric transmission of either linearly or circularly polarized waves only. Crown Copyright & 2014 Published by Elsevier B.V. All rights reserved.

Keywords: Planar chiral metamaterial Asymmetric transmission Dual bands

1. Introduction Metamaterials, periodical artificial media with subwavelength unit cell structures, are engineered to possess unusual electromagnetic properties not found in natural materials [1]. Over past few years, as one kind of metamaterial, the chiral metamaterial (CMM) has attracted extensive attention in many research groups [2]. A planar CMM does not exhibit any mirror symmetry, or it cannot be brought into congruence with its mirror image unless it is lifted off the substrate [3]. Owing to the chirality, a coupling between electric and magnetic fields exists. It leads to several novel properties such as strong optical activity [4,5], huge circular dichroism [6] and negative refractive index [7–9]. Due to their novel properties, planar CMMs can be applied to control the polarization states of the electromagnetic (EM) waves. Thus, they can be used as circular polarizers [10,11], polarization rotators [12] and polarization-independent absorbers [13]. Recently, lots of efforts have been focused on the asymmetric transmission (AT) in the chiral metamaterials [14–27], where the circular conversion dichroism [14,15] of EM waves greatly contributes to this interesting phenomenon. In contrast to the nonreciprocal Faraday effect that can also control the polarization state of EM wave, the AT phenomenon in the chiral metamaterial does not violate the reciprocal theory and requires no magnetic medium. Since Fedotov et al. [14] firstly proposed the AT phenomenon in the fish-scale planar chiral model, the AT of linearly

n

Corresponding author. E-mail address: [email protected] (J.-F. Dong).

http://dx.doi.org/10.1016/j.optcom.2014.02.053 0030-4018 Crown Copyright & 2014 Published by Elsevier B.V. All rights reserved.

and circularly polarized waves in CMMs was investigated by theoretical analysis, numerical simulation and experimental verification in microwave [14,17,18,20,22–26], THz [19] and optical [15,16,21] regions. However, most of the reported models can realize either linearly [22–26] or circularly [14–20] polarized ATs only. The CMM structures in which the ATs of both linearly and circularly polarized waves can be realized simultaneously are rarely reported. In the present paper, we propose a simple CMM structure, which consists of stacked pair of double split-ring resonators (DSRRs) placed on two sides of a substrate. It can achieve the dual-band linearly polarized ATs. Specifically, a theoretical model is applied to interpret the dual-band effect. In addition, due to the coupling between the SRRs, the designed planar structure can also achieve the ATs simultaneously for both linearly and circularly polarized waves, which have been previously reported only in the 3D chiral hybridized model [21].

2. Theory A linearly polarized wave is assumed to propagate through the chiral medium which is symmetrically embedded in vacuum. The relationship between the incident and transmitted waves can be related by the Jones matrix T [21] asEt ¼ TEi . For a linear polarization " # t xx t xy , where the indices f and basis, T can be expressed asT flin ¼ t yx t yy lin indicate that the wave is along the forward (þ z direction) and a special linear basis with basis vectors parallel to the coordinate

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D.-Y. Liu et al. / Optics Communications 323 (2014) 19–22

axes, respectively. The first and second subscripts x or y in matrix elements indicate the polarization states of the transmitted and incident waves, respectively. Similarly, for a circular polarization " # tþ þ tþ  f basis, T can be written asT cir ¼ , where indicesþand t þ t   denote the right and left circular polarized waves propagating along the þz direction, respectively. The transform of the basis vectors from linear to circular polarization states results in T fcir ¼

tþ þ t þ

tþ  t 

! ¼

1 t xx þ t yy þ iðt xy  t yx Þ 2 t xx  t yy þ iðt xy þ t yx Þ

t xx  t yy  iðt xy þ t yx Þ t xx þ t yy  iðt xy  t yx Þ

!

ð1Þ

The AT parameter Δ [21] is usually defined as the difference between the transmittances in the two opposite propagation directions (þ z and z) for a certain polarization state. For the linearly polarized wave, the Δ can be defined as

Δxlin ¼ jt yx j2  jt xy j2 ¼  Δylin For the circularly polarized wave,

ð2Þ

Δ should be

þ  Δcir ¼ jt  þ j2  jt þ  j2 ¼  Δcir

ð3Þ

According to the above equations, the AT for both linear and circular polarizations (Δlin a0 and Δcir a0) can be obtained as long as the four elements of the Jones matrix are not equal to each other [21]. To understand the physical mechanism of dual-band effect in this chiral structure better, a particle model (PM) [22,28,29] is adopted. When the EM wave illuminates the chiral structure, each layer of the metallic SRRs can be viewed as two orthogonal separated charge harmonic oscillators moving along the x and y directions with an effective displacement of dx and dy, respectively. It is noteworthy that the coupling between the two metallic layers should be taken into account. Thus, the equations for PM can be

Fig. 1. Geometry of the unit cell of designed chiral structure: (a) the top layer with two twisted SRRs, (b) the bottom layer, (c) the cross-section, and (d) the perspective drawing and the Cartesian coordinate system.

For the incoming plane wave polarized parallel to the x-axis, Eq. (4) can be solved, and txx, tyx can be found as t xx p dxB and t yx p dyB For polarization in y direction, we can get t xy p dxB and t yy p dyB

: t yx p fc4 ðω2yA  ω2  iωγ yA Þ þ c5 ðω2yA  ω2  iωγ yA Þðω2xB  ω2 iωγ xB Þ þ c6 ðω2xB  ω2  iωγ xB Þ þ f 2 ðsui vj Þg

8 2 ∂ dxA qxA > ∂ 2 > > > ∂t 2 þ γ xA ∂tdxA þ ωxA dxA þ syA xA dyA þ syB xA dyB þ sxB xA dxB ¼ mxA ExA > > > 2 > ∂ dxB qxB ∂ 2 > > < ∂t 2 þ γ xB ∂t dxB þ ωxB dxB þ syA xB dyA þ syB xB dyB þ sxA xB dxA ¼ m B ExB x

2

∂ dyA qA > þ γ yA ∂t∂ dyA þ ω2yA dyA þ syB yA dyB þ sxB yA dxB þ sxA yA dxA ¼ my A EyA > > ∂t 2 > y > > > ∂2 d B > qyB y > ∂ 2 > : ∂t 2 þ γ yB ∂t dyB þ ωyB dyB þ syA yB dyA þ sxB yB dxB þ sxA yB dxA ¼ myB EyB

ð4Þ

where A, B correspond to the upper and bottom of the metallic layer (shown in Fig. 1(a) and (b)), respectively. The four parameters xA, yA, xB and yB are represented by ui or vj (u; v ¼ x; yand i; j ¼ A; B). Eui denotes the external electric field imposed on the oscillators and its polarized direction is along u direction with the corresponding displacement dui . The parameters mui and qui are the effective mass and charge of the harmonic oscillators, respectively, while γ ui and ωui stand for damping of the harmonic oscillators and resonant angular frequency of ui, respectively. sui vj is the coupling strength between the two oscillators of ui and vj. It should be noted that the coupling between them is taken into consideration in the equations, when each metallic layer composed of two coaxial SRRs is viewed as a unit.

ð6Þ

It is not easy to solve Eq. (4); hence we just explore the solutions in Fourier domain qualitatively as follows:

8 < t xy p fc1 ðω2xA  ω2  iwγ xA Þ þ c2 ðω2xA  ω2  iwγ xA Þðω2yB  ω2  iwγ yB Þ þ c3 ðω2yB  ω2  iwγ yB Þ þ f 1 ðsui vj Þg

expressed as follows:

ð5Þ

ð7Þ

where cn (n ¼1,2,……,6) are constants and f k ðsui vj Þ (k¼ 1,2), independent of frequency, are functions of sui vj while ω is the angular frequency of the incident EM wave. From Eq. (7), txy is mainly controlled by the resonant angular frequencies ω2xA and ω2yB with their corresponding damping coefficients γ xA andγ yB , respectively, while tyx is dominated by ω2yA ,γ yA ,ω2xB and γ xB , respectively. The peak values for txy and tyx at different angular frequencies can be obtained, as long as these parameters have different values. In what follows, as a good example, the designed chiral structure is presented to demonstrate the above analysis' results.

3. Numerical simulations and results Fig. 1 shows the unit cell of the proposed CMM, where the SRRs are chosen, since the CMM structure composed of SRRs is a good candidate for achieving AT [4,23–27]. This structure is composed of a dielectric substrate made by Rogers TMM 4 sandwiched between two copper layers (Fig. 1(c)). The relative permittivity of the substrate is 4.5 with a dielectric loss tangent of 0.002, and the substrate thickness is ts ¼1.4 mm. The SRRs (Fig. 1(a) and (b)) are made of copper; for the copper layer, the thickness is tc ¼ 32 μm

D.-Y. Liu et al. / Optics Communications 323 (2014) 19–22

1.0

1.0 txx txy tyx tyy

0.6

txx txy tyx tyy

0.8

Transmission

Transmission

0.8

0.4

0.6 0.4 0.2

0.2

0.0

21

8

10

12

14

16

0.0

18

8

10

Frequency(GHz)

12

14

16

18

Frequency(GHz)

Fig. 2. The four elements of transmission coefficients of the designed CMM in the  z (a) and þz (b) directions.

1.0

1.0

0.8

0.8

PM FDTD

0.6

PM FDTD

txy

tyx

0.6

0.4

0.4

0.2

0.2

0.0

8

9

10

11

12

13

0.0

12

13

Frequency (GHz)

14

15

16

17

Frequency (GHz)

Fig. 3. The rigorously calculated (FDTD) cross-polarized transmissions tyx (a) and txy (b) compared with results of the PM.

and the conductivity is 5.8  107 S/m. In the designed chiral structure, the first SRR-layer on one side of substrate contains two coaxial SRRs twisted by 901, shown in Fig. 1(a); the second SRR-layer on the other side of substrate also contains the same two SRRs (Fig. 1 (b)), and the second layer has a rotational degree of 901 with respect to the first SRR-layer, shown in Fig. 1(d). The geometrical parameters of the unit cell are as follows: ax ¼ay ¼9 mm, r1 ¼ 4 mm, r2 ¼2.4 mm, w¼0.8 mm, and g¼ 0.2 mm. Besides, the two SRRs have the same width w and split gap g. The operating frequency is 8.9 GHz and 16.8 GHz; hence this proposed structure is electrically thin, since (ts þ2tc)/λ is 0.043 and 0.082 for 8.9 GHz and 16.8 GHz, respectively. In addition, the period in x and y direction is also electrically small since ax/λ is 0.27 at 8.9 GHz and 0.53 at 16.8 GHz. We adopted the finite difference time domain (FDTD) method to start the analysis of the EM transmission properties of the proposed CMM. The boundaries are selected to be periodic in x–y plane while open in the z direction. In order to characterize the response of the CMM, the normal incident linearly polarized EM wave over a broad frequency range was employed to illuminate the unit cell along the  z direction. It should be noted that for an x-polarized incident wave, linear transmission coefficients txx and tyx can be obtained. Similarly, the tyy and txy are also obtained when the incident wave is y-polarized. The numerical linear transmission coefficients are illustrated in Fig. 2. Fig. 2(a) reveals that this CMM structure creates an orthogonal polarized component of the incident wave, since the two coupling components txy and tyx are not equal to zero. In contrast to several bi-layered chiral structures whose co-polarized terms are equal [22,23,25–27], the co-polarization transmission coefficients txx and tyy of this designed CMM are not identical in certain frequency regions (from 8.7 GHz to 12.8 GHz and from 16 GHz to 18.5 GHz). The cross-polarization transmission term txy

Table 1 The PM parameters for fitting. Parameters

tyx

txy

ωyA ,ωxB ;ωxA ,ωyB [rad/s] γ yA ,γ xB ;γ xA ,γ yB [Hz] c1 ,c2 ,c3 ;c4 ,c5 ,c6

7.88  1010, 5.59  1010 6.80  109, 2.56  109  2.99

10.70  1010, 7.79  1010 9.10  109, 18.80  109  2.34

Note: c1 ¼ c2 ¼ c3 , c4 ¼ c5 ¼ c6

is in a sharp contrast to tyx across the whole frequency region. Interestingly, there are two pass bands for tyx; their peaks can reach 0.82 and 0.43 at the frequency of 8.9 GHz and 12.2 GHz, respectively. Similarity, txy also has two resonant peaks, one reaches 0.87 around 16.7 GHz and the other is 0.35 at 12.7 GHz. The co-polarization txx and tyy are below 0.05 and 0.3 at the resonant frequencies of 8.9 GHz and 16.7 GHz, respectively. It indicates that the x-polarized or y-polarized wave converts mostly to its orthogonal component. Fig. 2(a) and (b) shows that txy and tyx interchange when the propagation direction is reversed, while the values of txx and tyy remain unchanged. In order to confirm the validity of the PM proposed above, we performed the numerical FDTD simulations of cross-polarized transmission again, and compared them with results obtained by the PM in Fig. 3. The PM parameters are used to fit the results of FDTD method, listed in Table 1. The model-predicted crosspolarized transmission spectra of tyx and txy are compared with numerical counterparts, shown in Fig. 3(a) and (b), respectively. Clearly, the curves for both tyx and txy are in good qualitative agreement for the rigorous FDTD results (solid line) and the PM (dashed dotted line).

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for circular polarization

0.6

x y

0.4 0.2 0.0 -0.2 -0.4

(+,-)

x,y for linear polarization

0.8

-0.6

0.1 +

0.0

-0.1

-0.8 8

10

12

14

16

18

8

Frequency(GHz)

10

12

14

16

18

Frequency(GHz)

Fig. 4. AT parameter Δ for linear (a) and circular (b) polarization states; the polarization states (c) of the transmitted wave at different frequencies when the incident wave is x-polarized; the direction of polarized states is indicated by the black arrow.

To confirm the presence of AT phenomenon in this designed structure, we calculated the AT parameter Δ for linear and circular polarizations shown in Fig. 4. As illustrated in Fig. 4(a), both AT parameters Δx and Δy exhibit two opposite peaks that can reach a maximum of 70.67 and 7 0.78 around the frequency of 8.9 GHz and 16.7 GHz, respectively. It is noted that no AT effect of circular polarization wave was found for the bi-layered CMMs in previous reports [22,23,25–27]. However, Fig. 4(b) exactly demonstrates that this designed structure possesses the ability to achieve the AT of circularly polarized wave, since the values of Δ 7 reach up to 14% at the resonant frequency of 17 GHz. In order to explore the coupling effect that can convert x- or ypolarized wave into the orthogonal component (y- or x-polarized), we obtain the polarization states of transmitted wave by calculating the corresponding ellipticity and polarization azimuth (not show here) of transmitted wave at different frequencies as shown in Fig. 4(c). It can be clearly observed that when the incident wave is x-polarized, the transmitted wave varies from linear to ellipse (þ ,  ) polarization states. It should be noted that the circularly polarized state cannot be realized, since the ellipticity cannot reach 451 across the whole simulated frequency range. 4. Conclusion In summary, we have proposed a chiral structure composed of double twisted split-rings placed on both sides of a dielectric substrate. The chiral metamaterial has been theoretically and numerically revealed to achieve dual-band AT of linearly polarized waves. In addition, the designed chiral structure can realize the AT for both linear and circular polarization states of EM waves. This designed chiral structure can also manipulate the polarization states of EM waves. It is predicted that this chiral structure can achieve the same functionality in other frequency bands, such as THz and optical regions by scaling down to a proper scale. Acknowledgments This work is supported by the National Natural Science Foundation of China (61078060), and Ningbo Optoelectronic Materials

and Devices Creative Team (2009B21007), and is partially sponsored by K.C. Wong Magna Fund in Ningbo University.

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