Left Handed Metamaterial with = – 0 and = – 0 and Some Applications

Yongjun Huang, Guangjun Wen, Tianqian Li School of Communication and Information Engineering

University of Electronic Science and Technology of China Chengdu, China

yongjunh@uestc.edu.cn

Kang Xie School of Opto-Electronic Information

University of Electronic Science and Technology of China Chengdu, China

Abstract—A left handed metamaterial (LHM) with = – 0 and = – 0 is proposed and its properties are theoretically and numerically analyzed. The LHM is composed of ferrites to provide a negative permeability and a wire array to provide a negative permittivity. The parameter conditions of satisfying aforementioned properties are investigated. The analyzed results show that the LHM has a nearly perfect negative pass band with very small reflections and losses in the microwave frequencies and the negative pass band can be tuned by changing the applied magnetic field. This LHM can be applied to various areas such as microwave engineering areas. In this paper, the applications of the LHM including the microwave filter and phase shifter are discussed respectively.

Keywords-left handed metamaterial; ferrites; applications; filter; phase shifter

I. INTRODUCTION Since the left handed metamaterial (LHM) predicted by

Veselago [1] in 1968 was experimentally realized by Smith, et al. in 2000 [2] and verified by Shelby, et al. [3] in 2001 through negative refraction in a prism sample, much attention has been attracted on designing various LHMs [4-6] and investigating properties [7] and applications [8]. Much of the fascination in LHMs arises from their unusual electromagnetic properties such as the reversals of both Doppler shift and Cherenkov radiation [1], enhancement of evanescent wave, and sub-wavelength resolution imaging [9], etc. In the various LHMs, most of the them are realized by artificial metallic structures with metallic plasma resonance such as using wires to produce effective negative permittivity and using split-ring resonators (SRR) to provide effective negative permeability [2-4]. Recently, the investigations of electromagnetic cloak of invisibility based on the metallic plasma resonance structure metamaterial [10] have generated great interests. However, most LHMs proposed to date are based on immutable structure of the unit cell and result in a narrow band or not at all tunable.

Recently, some researchers have proposed other LHMs consisting of ferrites and wires, that is, substitute ferrites for the SRR structure to obtain the negative effective parameters [11,12]. The ferrites based LHM possesses some properties such as broad negative pass-band, low-loss, and tunability. In this paper, based on the LHM consisting of ferrites and wires, we propose and analyze the tunable LHM with = – 0 and = – 0. The effective parameters, the EM wave transmission

properties, and some applications are theoretically and numerically investigated in this paper.

II. MODEL AND THEORIES The single unit cell of the ferrites based LHM is shown in

Fig. 1. The conducted regions are within the circles of radius r1. The region between r1 and r2 is filled with a nonmagnetic dielectric insulator. The region surrounding the cladding wires is filled with ferrites. The lattice spacing a is much larger than r1 with r2 chosen such that r2 (r1·a)1/2. An EM wave propagates along the y axis with the electronic field along the z axis and the magnetic field along the x axis. And a dc applied magnetic field acts on the ferrites along the z axis. So the effective permeability and permittivity can be obtained from the following expresses [11]

2 2

0

20

0

( ) ( )

( ) ( )

Seff

S

H M

H H M

ωμ μ γ

ωμμ γ

+ −=

+ −, (1)

00 0

4

S

H H iM

ω πμ γ μ γ

Λ= − , (2)

Figure 1. Single unit cell of the LHM (left) and the top view (right).

This work was supported by National Natural Science Foundation ofChina under Grant Nos. 60571024, 60771046, and 60588502.

Proceedings of 2009 IEEE International Conference onApplied Superconductivity and Electromagnetic DevicesChengdu, China, September 25-27, 2009

ID1064

978-1-4244-3687-3/09/$25.00 ©2009 IEEE 119

20 0 2

0 01 2

[ ( )( ln {ln 1.602})]2

eff f eff

effeff

a r air r

ε ε σω σε ε

ωε μ μπ

= −+ + −

(3)

Here MS is the saturation magnetization of ferrites. H0 is the applied magnetic field. is the gyromagnetic ratio. is the phenomenological damping parameter describing losses intrinsic to the magnetic material [11]. f is the permittivity ferrite. eff is the effective conductivity of wire arrays. 0 is the permeability of air. is the angular frequency.

So one can use these theoretical results mentioned above to design the LHM so that both the effective permeability and permittivity are approximate to -1 in a broad frequency band and the losses in such medium are very small. First the effective permeability can be easily calculated from (1) and (2) with appropriate values. Here we design this LHM in X-band (8.2 GHz – 12.4 GHz). So the following parameter values are chosen and the calculated result of the permeability is shown in Fig. 2 (red solid line). The permittivity of the ferrites f = 4 0, magnetization Ms = 3.5 × 105 A/m, and applied magnetic field H = 4 × 104 A/m.

However the effective permittivity is difficult to decision due to the complexity of (3). Here we use the MATLAB software to calculate and optimize the parameters so that the effective permittivity equal to -1. The optimized parameter values are shown in the following. The lattice constant a = 3.6 × 10-3 m, radii of each copper wire r1 = 2 × 10-5 m, and radii of cladding r2 = 4.8 × 10-4 m. The calculated effective permittivity is shown in Fig. 2 (green dashed line).

2 4 6 8 10 12 14 16-60

-40

-20

0

20

40

60

Frequency (GHz)

9 9.5 10 10.5 11-4

-3

-2

-1

0

1

eff

μef

fε

Re( )effεIm( )effε

Im( )effμRe( )effμ

Figure 2. The calculated effective parameters (permeability and permittivity) of the LHM. The inset figure is the magnification in the frequency range of 9 GHz – 11 GHz.

From Fig. 2 it can be known that the effective permeability of the ferrites exhibits negative values in a broad frequency band (4.4 GHz – 13.7 GHz) and the effective permittivity of the wire array embedded in the ferrites host with cladding exhibits negative values in the frequency range of 7.5 GHz – 10.6 GHz. So one can expect that the composite metamaterial exhibits negative refractive index in the range of both the effective permeability and permittivity are negative (7.5 GHz – 10.6 GHz). From Fig. 2 one can also know that the image parts of the two parameters are very small, resulting in low loss within the composite.

And the most significant character is that the real parts of the effective permeability and permittivity are equal to -1 and the image parts of the two parameters are approximate to 0 at 10 GHz (see the inset figure of Fig. 2). It means that the LHM is impedance matching to the air. Therefore the composite LHM can perfectly transfer the EM waves with very low losses and reflections at 10 GHz. These characters are the necessary conditions of designing the perfect lens, frequency selective surface, cloak, and other microwave devices. These applications will be discussed in the next section.

The tunability of the LHM presented here is due to the tunability of the ferrites host under different applied magnetic fields [11,12]. Figure 3 shows the tunability of the ferrites host under the applied magnetic fields ranged from 40000 A/m to 70000 A/m. It can be known that the negative permeability of the ferrites host shift from low frequency to high frequency by changing the applied magnetic fields. Therefore the frequency points of the effective permeability equal to -1 are shift from 10 GHz to 1.6 GHz (This result is not shown clearly in Fig. 3). So the LHM presented in this paper can used to design the perfect lens [9] and microwave devices in arbitrary frequencies.

2 4 6 8 10 12 14 16-60

-40

-20

0

20

40

60

Frequency (GHz)

eff

μ

Re( )effμ

Im( )effμ

Figure 3. The theoretical demonstrated tunability of the LHM consisting ferrites host and wire array.

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III. NUMERICAL RESULTS In this section, we numerically investigate the LHM by

using commercial software (HFSS) with the parameters presented in section II. To investigate the transmission properties of the LHM in a broad frequency band (2 GHz – 16 GHz), a planar waveguide system with a cross section of 36 × 8 mm2 is used. The cutoff frequency of first high-order mode of the planar waveguide is fc = c/2d 19 GHz. So this system can transfer TEM wave only in the range from dc to 19 GHz. The LHM slab is put at the middle of the planar waveguide and the two side walls are Master and Slave boundaries. Fig. 4 shows the transmission parameters of the LHM slab under the applied magnetic of H = 4 × 104 A/m as a function of frequency.

2 4 6 8 10 12 14 16-300

-250

-200

-150

-100

-50

0

Frequency (GHz)

Tra

nsm

issi

on p

aram

eter

s (d

B)

(a)

S21

S11

(a)

8 9 10 11 12 13 14-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency (GHz)

Tra

nsm

issi

on p

aram

eter

s (d

B)

(b)

S21

S11

(b)

Figure 4. (a) Numerically demonstrated transmission properties of the LHM under the applied magnetic field of H = 4 × 104 A/m. (b) The magnification of the transmission parameters in the frequency range of 8 GHz – 14 GHz.

It clearly shows that the ferrites based LHM can transfer the EM waves with very low losses and reflections in the range of 9.55 GHz – 11.85 GHz. The pass band where the loss is smaller than 4 dB is 2.3 GHz centered at 10.653 GHz. The transmission peak of S21 in the pass band is –1.6 dB at 10.6 GHz and the deep value of the S11 is –78 dB at 10.4 GHz. Therefore one can expect that the LHM presented here exhibits a perfect negative pass band near the 10.6 GHz with the character of = – 0 and = – 0. From Fig. 4(a) one can know that there is another pass-band at 3.8 GHz. And there are some differences between the theoretical results and numerical results. One can interpret these results as follows. In Fig. 3 the effective permeability and permittivity are both positive in the range of 3.2 GHz --- 4.4 GHz, so the EM waves can also transferred in such frequency band with a positive refraction character. The properties of = – 0 and = – 0 is occurred at 10 GHz in theoretically while the perfect negative frequency is occurred at 10.5 GHz in numerically. At the same time the negative pass band of the LHM is occurred at 7.5 GHz – 10.6 GHz in theoretically and occurred at 9.55 GHz – 11. 85 GHz in numerically. This is due to the difference applied magnetic filed in simulation and theory.

IV. SOME APPLICATIONS In the end, we show some applications of the LHM

presented in this paper. Because of the novel properties and the controllability of the parameters, LHM has the applications that other conventional materials do not possess. We mainly discuss two microwave engineering applications in this paper. The first application is that the LHM can be used to fabricate microwave band-pass filter. As shown in Fig. 4, The LHM can transfer the EM waves with very low loss and reflection in the frequency band of 9.55 GHz – 10.6 GHz and forbid mostly EM waves under 9.55 GHz and above 10.6 GHz. The in-band insertion loss is less than 4 dB and the return loss is greater than 30 dB. Moreover, the pass band can also be tuned by changing the applied magnetic field (see Fig. 3). So the LHM band-pass filter can be fabricated in arbitrary frequencies from low frequency to the millimeter wave and optical frequencies.

The second application is that the LHM presented here can be used to fabricate microwave linear phase shifter due to the negative phase velocity. Fig. 5 shows the phase response and S21 of the LHM under the applied magnetic field of H = 4 × 104 A/m in the negative pass band of 9.55 GHz – 11.85 GHz. It clearly shows that the LHM exhibits a maximum insertion loss of 4.2 dB, a minimum of 1.6 dB, and a minimum return loss of 24.5 dB [see Fig. 4(b)] across all phase states over 9.55 GHz to 11.85 GHz. And the phase slope is about 80°/GHz and the maximum deviation of phase from linear is just ± 10°.

So the LHM presented in this paper can be used widely in microwave engineering applications. The LHM can also be used in other application areas such as the perfect lens, cloak, and absorber [9,10], leak wave antenna [14], coupler [15] and power divider [16]. It just changes the parameter values of the LHM in different applications.

Our next work will mainly focus on designing and fabricating the metamaterial loaded band-pass filter and the microwave linear phase shifter in the X-band by using the ferrites and wire array. The mostly big challenge is the

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manufacturing process of the ferrites based metamaterial. We will fabricate the metamaterial by using a sample way that is using ferrites slabs and PCBs that has been reported in [13]. We have fabricate the metamaterial consisting of the ferrites and PCBs in the X-band and experimentally measured the transmission character. These works will help us more easily to design and fabricate the metamaterial loaded filter and the phase shifter.

10 10.5 11 11.5-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

S21

(dB

)

9.55 10 10.5 11 11.5 11.850

20

40

60

80

100

120

140

160

180

200

Frequency (GHz)

Pha

se (

degr

ee)

Phase

S21

Figure 5. The transmission S21 and the Phase response of the LHM under the applied magnetic field of H = 4 × 104 A/m in the negative pass band (9.55 GHz – 11.85 GHz).

V. CONCLUSION In this paper, a left handed metamaterial (LHM) with = ---

0 and = --- 0 is proposed and its properties are theoretically and numerically analyzed. Both the theoretical and numerical results show that the LHM has a negative pass-band in the microwave frequency band with a very small reflections and losses. The LHM can also be tuned by changing the applied magnetic fields. The LHM has wide applications in the microwave engineering areas and other areas. The microwave band-pass filter and linear phase shifter are discussed, respectively.

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