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Optik 124 (2013) 1683– 1685

Contents lists available at SciVerse ScienceDirect

Optik

jou rn al homepage: www.elsev ier .de / i j leo

flat-top filter based on a tri-layer structure with anisotropic single-negativeetamaterial

ingting Tang ∗, Xiuying Gao, Xiujun He, Wenli Liuollege of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China

a r t i c l e i n f o

rticle history:eceived 8 December 2011ccepted 10 May 2012

a b s t r a c t

We construct a flat-top filter by use of a tri-layer structure with AENG and AMNG metamaterials. Transfer-matrix method is used to calculate the transmittance of our tri-layer structure, and results show that twotunneling modes emerge at some frequencies and then will merge with each other, thus a flat-top filter

eywords:nisotropicetamaterial

lat-top

is realized. We also show that metamaterial loss significantly reduces the transparent light, but does notchange the transmittance distribution of light.

© 2012 Elsevier GmbH. All rights reserved.

ilter

. Introduction

As a new type of artificial composite, metamaterial hasttracted much attention recently. It possesses a large numberf unusual electromagnetic properties, such as negative refrac-ive index, antiparallel group and phase velocities [1–3]. The

etamaterials include double-negative (DNG) and single-negativeSNG) material. It can be used to realize new-style photonicevices, including compact directional coupler (DC) [4], polar-

zation splitter (PS) [5] and filters with adjustable pass band.s typical metamaterials with epsilon-negative (ENG) or mu-egative (MNG) are realized by stacking sheet of substrate withrinted split-ring resonators (SRRs) on their surface [6] or embed-ed thin metal wires [7], and they always exhibit anisotropicroperty and have tensor material parameters. Thus, it will beore realistic to analyze photonic devices containing anisotropic

psilon-negative (AENG) and anisotropic mu-negative (AMNG)etamaterials.In this paper, we construct a flat-top filter by use of a

ri-layer structure with AENG and AMNG metamaterials. Transfer-atrix method is used to calculate the transmittance of our

ri-layer structure, and results shows that two tunneling modesmerge at some frequencies and then will merge with each

ther. In addition, the effect of metamaterial loss is alsoonsidered.

∗ Corresponding author.E-mail address: skottt@163.com (T. Tang).

030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2012.06.008

2. Theory analysis and modeling

We consider a tri-layer structure created by uniaxial AENG(black area A) and AMNG (gray area B) material as shown in Fig. 1.

The anisotropic metamaterials are characterized by:

�ε1 = ε0

⎛⎜⎝

ε‖ 0 0

0 ε‖ 0

0 0 ε⊥

⎞⎟⎠

and relative permeability �1 in layer 1 and relative permittivity ε2and

��2 = �0

⎛⎜⎝

�‖ 0 0

0 �‖ 0

0 0 �⊥

⎞⎟⎠

in layer 2, respectively. In the same layer, the electric and magneticfields at any two positions z and z + �z can be related to each otherby a transfer matrix [8]:

Mj(�z) =(

cos(kjz�z) i sin(kj

z�z)/qj

iqj sin(kjz�z) cos(kj

z�z)

), j = (1, 2) (1)

where kx = k0sin� and � is the incidence angle with +z direction. For√ √

TE wave, kz1 = ε‖�1k2

0 − k2x , kz2 = (ε2�‖k2

0 − �‖k2x )/�⊥,

q1 = kz1/�1k0 and q2 = kz2/�‖k0. For TM wave, kz1 =√ε‖�1k2

0 − ε‖k2x /ε⊥, kz2 =

√ε2�‖k2

0 − k2x , q1 = kz1/ε‖k0 and

q2 = kz2/ε2k0.

1684 T. Tang et al. / Optik 124 (2013) 1683– 1685

n2 n1

d2 d1

x

zo

d2

θ n2

Laye r2 Laye r2Layer1

t

t

w

Mm

3

tw

agd

ε

�

mr�Bf

F

teta

m

ε

wnrωε

302826242220181614

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

issta

nce

d =2 mm

d =3 mm

d =4.5 mm

All aforementioned theoretical and numerical analyses arebased on an assumption that all the materials are lossless, but itis well known that most of metamaterials have complex refractiveindices whose imaginary parts are unnegligible. So it is necessary

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

itta

nce

Fig. 1. A tri-layer structure created by uniaxial AENG and AMNG metamaterial.

Then, the transmission coefficient t can be obtained from theransfer matrix:

= 2p

(M11 + M12p)p + M21 + M22p(2)

here Mij(i, j = 1,2) is the matrix elements of (MbMaMb). Here

a = M1(d1), Mb = M2(d2) and p =√

k20 − k2

x /k0. The power trans-ittance is determined by T = tt∗.

. Simulation results and discussions

Now we study transparent modes in our tri-layer structure. Inhis paper, we mainly focus on TE waves, and the solutions for TMaves can also be derived by the same method.

In our structure the effective permittivity and permeability ofnisotropic metamaterial can be derived as Ref. [9], and then theeneral zero (volume) average permittivity and permeability con-itions for TE wave of the 1DPC can be written as:

= (ε‖ − sin2 �/�1)d1 + 2ε2d2

d1 + 2d2= 0 (3)

= �1d1 + 2[�‖(1 − sin2�/ε2�⊥)]d2

d1 + 2d2= 0 (4)

Eqs. (5) and (6) can be regarded as the condition of tunnelingodes merging. Without loss of generality, we suppose dispersion

elations of metamaterials are ε‖ = f (ω), ε⊥ = g(ω), �‖ = h(ω) and⊥ = k(ω), while �1and ε2 are constants independent of frequency.y substitution of these variables into Eqs. (5) and (6), the merging

requency can be obtained by solving the following equation:

(ω) = �1d1 + 2h(ω)d2

− [h(ω)d2�1 · (d1f (ω) + 2ε2d2)]/ε2k(ω)d1 = 0 (5)

If F(ω) is close to zero, a tunneling mode will emerge. Then if twounneling modes are getting close to each other, they will mergeventually at some frequencies. Because of special dispersion rela-ion, there may be more than one tunneling mode. When F(ωp) ( 0,

perfect tunneling mode is formed at ω = ωp.The permittivity of AENG media and permeability of AMNG

edia are given by Drude model [10]:

⊥ = 1 − ω2ev

ω2, ε‖ = 1 − ω2

eh

ω2, �⊥ = 1 − ω2

mv

ω2, �‖ = 1 − ω2

mh

ω2

(6)

here ωeh (ωmh) and ωev (ωmv) are the effective electric (mag-

etic) plasma frequencies in the horizontal and vertical directions,espectively. Here ω is in unit of gigahertz, and we choose �1 = 1,ev = 34.64 GHz and ωeh = 31.3 GHz for the AENG material. Also

2 = 1, ωmv = 31.62 GHz and ωmh = 30 GHz for the AMNG material.

Frequ ency (GH z)

Fig. 2. Transmittance of the tri-layer structure when d1 is different and � = 61◦ .

In what following, we will show the tunneling modes of ourtri-layer structure and the generation of a flat-top filter. We firstassume that d2 = 2 mm and � = 61◦, the transmittance at differ-ent frequency of our tri-layer structure is shown in Fig. 2. Whend1 = 2 mm, there are two tunneling modes, and with the increaseof d1, they will get close to each other, and then when d1 = 4.5 mm,they merge into one broad pass band. Thus a flat-top filter is real-ized.

Then we fix d1 = 4.5 mm and d2 = 2 mm to show the influence ofincident angle on transmittance of our filter in Fig. 3. When � = 75◦,there are also two tunneling modes and they are very close to eachother. As � decreases, the frequency gap between the two tunnelingmodes becomes smaller. When � = 61◦, a broad band flat top in thetransmittance is observed.

Based on the above analysis and simulation, we constructanother tri-layer structure by changing the location of AENG andAMNG metamaterial as shown in Fig. 4.

We first assume that d1 = 2 mm and � = 34◦, the transmittanceat different frequency of our tri-layer structure is shown in Fig. 5.When d2 = 2 mm, there are also two tunneling modes, and they aregetting close to each other when d2 is increasing. At last, they mergeinto a flat-top pass band when d2 = 4.5 mm.

Then we fix d1 = 2 mm and d2 = 4.5 mm to show the influenceof incident angle on transmittance of the newly constructed tri-layer structure in Fig. 6. When � = 62◦, there are also two tunnelingmodes and they are very close to each other. As � decreases, thefrequency gap between the two tunneling modes becomes smaller.When � = 34◦, a broad band flat top in the transmittance is observed.

25242322212019

Frequency (GHz)

Fig. 3. Transmittance of the tri-layer structure when d1 = 4.5 mm and d2 = 2 mm.

T. Tang et al. / Optik 124 (2013) 1683– 1685 1685

n1

d1

x

zo

d2

θ n2

Layer1Layer1 Layer2

n1

d1

Fig. 4. A tri-layer structure created by exchanging the location of uniaxial AENG andAMNG metamaterial.

32302826242220181614

0.0

0.2

0.4

0.6

0.8

1.0

Frequency (GHz)

Tra

nsm

itta

nce

Fig. 5. Transmittance of the tri-layer structure when d1 is different and � = 34◦ .

2624222018

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

itta

nce

F

tct

wt�ω

32302826242220181614

0.0

0.1

0.2

0.3

0.4

0.5

Tra

nsm

itta

nce

Frequency (GHz)

ig. 6. Transmittance of the tri-layer structure when d1 = 2 mm and d2 = 4.5 mm.

o have a discussion about the effect of material loss in our filteraused by SNG material. This effect can be taken into account byaking for the SNG material the relative permittivity:

ε⊥ = 1 − ω2ev

ω2 − iωıe, ε‖ = 1 − ω2

eh

ω2 − iωıe,

�⊥ = 1 − ω2mv

ω2 − iωım, �‖ = 1 − ω2

mh

ω2 − iωım(8)

here ıe is the electronic damping frequency and ım ishe magnetic damping frequency. Here we also choose

1 = 1, ωev = 34.64 GHz, ωeh = 31.3 GHz, ε2 = 1, ωmv = 31.62 GHz,mh = 31.30 GHz, d1 = 2 mm, d2 = 4.5 mm, � = 34◦ and ıe = ım = ı = 1

[

Frequency (GHz)

Fig. 7. Transmittance of the tri-layer structure when ıe = ım = ı = 1.

(for simplicity), the transmittance of the filters shown in Fig. 7. Wecan find that the loss induced by the imaginary part of permeabilityand permittivity significantly reduces the transmittance of light.Even ı is as small as 1, the flat-top of the filter totally disappears.Other simulation results show that when ı = 2, almost no light cantransmit through the tri-layer structure. But we also notice thatalthough huge reduction of transmittance of light is introduced byloss metamaterial, the distribution of light at different frequencyis never changed.

4. Conclusion

In this paper, we construct a flat-top filter by use of a tri-layerstructure with AENG and AMNG metamaterials. Transfer-matrixmethod is used to calculate the transmittance of our tri-layer struc-ture, and results show that two tunneling modes emerge at somefrequencies and then will merge with each other, and a flat-top fil-ter is realized. In addition, metamaterial loss significantly reducesthe transparent light, but does not change the transmittance distri-bution of light.

Acknowledgment

This work was supported by Project (KYTZ201122), supportedby the Scientific Research Foundation of CUIT.

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