June 2011, 18(3): 99–104 www.sciencedirect.com/science/journal/10058885 http://www.jcupt.com

The Journal of China Universities of Posts and Telecommunications

Design and analysis of quantized controllable doped left-handed materials

LIU Wen1, ZHANG Hong-xin1 ( ), WANG Hong1, LÜ Ying-hua1, CHEN Nan2, MEN Shuo3

1. School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China

2. Beijing National Railway Research and Design Institute of Signal and Communication, Beijing 100073, China

3. College of Electronic Information and Automation, Chongqing University of Technology, Chongqing 400054, China

Abstract

Two kinds of controllable doped left-handed materials (DLHMs) were designed by inserting inductors and capacitors into the traditional left-handed material (LHM) as heterogeneous elements respectively, which are DLHM with inductors (LDLHM) and DLHM with capacitors (CDLHM). The characteristics of transmission spectrum were studied by using finite-difference time-domain method (FDTD). Compared with the traditional LHM, the resonance strength of the LDLHM is weakened and the pass-band is narrowed, but with the increase of the value of the inserted inductors, the bandwidth is expanded. As capacitors inserted into the LHM, the pass-band of the CDLHM is expanded, but the pass-band is shifted to low frequency and the bandwidth is narrowed with the increase of the value of the capacitors, meanwhile, a new generated pass-band is also shifted to low frequency. Therefore, a quantized controllable doped left-handed material can be achieved.

Keywords electromagnetics, left-handed material, pass-band, capacitors, inductors

1 Introduction

In recent years, left-handed materials (LHMs) become more popular in the fields of electromagnetic (EM) and microwave because of their actual realization as metamaterials [1–2]. Science magazine even named LHMs as one of the ten scientific breakthroughs in 2003. It has been demonstrated that a thin continuous metallic wire array could have a first forbidden band from 0 Hz up to a low plasma frequency in a GHz range, where the effective permittivity of the medium is negative [3]. On the other hand, an array of split-ring resonators (SRRs) can exhibit a negative effective permeability for frequencies close to the magnetic resonance frequency of these structures [4]. By combining these SRRs and thin wires, Smith gave the first experimental demonstration of LHMs in 2000 [5]. One can observe negative Snell’s law, reverse Doppler effect, and reverse

Received date: 06-09-2010 Corresponding author: ZHANG Hong-xin, E-mail: hongxinzhang@bupt.edu.cn DOI: 10.1016/S1005-8885(10)60070-2

Cerenkov radiation when electromagnetic waves propagate in the LHMs [6]. However, current LHMs are designed with few pass-bands and the controllability of these materials also needs to be improved.

The concept of quantized controllable doped left-handed materials is investigated as follows in this paper: changes of transmission spectrum can be achieved by controlling the value of the inductors and capacitors inserted into the LHMs; the transmission properties of the doped left-handed materials (DLHMs) are analyzed using DLHM with inductors (LDLHM) and DLHM with capacitors (CDLHM). Firstly, a numerical simulation on the SRR-only array is conducted by using finite-difference time-domain method (FDTD), and the simulation result is compared with Bayindir’s measurement [7]. Noting a good agreement of the simulation result with Bayindir’s measurement, the characteristics of transmission spectrum of DLHMs are studied by using FDTD. The simulation results show that LDLHM can make the pass-band narrow and the resonance is weakened compared with that of the traditional LHM,

100 The Journal of China Universities of Posts and Telecommunications 2011

whereas CDLHM can expand the pass-band, also with a new generated pass-band. But with the increase of the value of inductors or capacitors, the bandwidth of LDLHM is expanded and that of CDLHM is narrowed. Therefore, the controllability of traditional LHMs is improved.

2 Split ring resonators

We first perform a numerical simulation for the SRR array constructed by Mehmet Bayindir [7]. The parameters of the SRR structure are shown in Fig. 1(a). The parameters of the unit cell SRR are 0.33 mmd w t , and 3 mml . The thickness of the SRR is 30 m. Fig. 1(b) presents the negative metamaterial composed of periodically arranged boards that have SRRs embedded on them.

(a) The unit cell of the SRR

(b) Schematic drawing of the negative metamaterial

Fig. 1 Parameters of the SRR structure

Mehmet had measured the response with EM wave incident to the SRR structure [7], and Fig. 2(a) shows that the negative metamaterial exhibits a forbidden band between 8.7 GHz and 10.2 GHz.

Our numerical simulation is based on the FDTD method. In the FDTD simulation, unit cells are the same as those in the measurement. We have employed 25 unit cells along the propagation direction and assumed periodic boundary conditions on the both sides perpendicular to the propagation direction. Fig. 2(b) shows the calculated transmission spectrum of this negative metamaterial. The SRRs give rise to a forbidden band from 8.7 GHz to 9.4 GHz, which is in good agreement with Mehmet’s measurement.

(a) Measured transmission spectrum of the SRRs [7]

(b) Calculated transmission spectrum with FDTD

Fig. 2 Constructed by Mehmet Bayindir.

3 Doped left-handed materials

Two kinds of doped left-handed materials, which are LDLHM and CDLHM, are discussed here. The wire and the SRR arrays in all the following simulations share the same structure shown in Fig. 3.

(a) The unit cell of the SRR array

(b) Schematic drawing of the negative metamaterial

Fig. 3 The structure of LHM

Issue 3 LIU Wen, et al. / Design and analysis of quantized controllable doped left-handed materials 101

The parameters of the unit cell of the SRR array are 0.3 mmd w t and 5.1 mml . The wires and the

SRRs are set on the both sides of the dielectric board. The distance between every two wires is 3.3 mmr and the distance between every two SRRs is 2.1 mmn . The metallic wires are continuous ones and their width and thickness are 0.3 mm and 30 m, respectively. On the other hand, the SRRs are also 30 m thick. The dielectric constant of the boards is b 4.41 and all boards are 21.6 mm long

in the propagation direction. In the following simulations, EM waves propagate along

the X axis and the electric field polarization aligned along the Y axis, whereas the magnetic field polarization is parallel to the Z axis. The perfectly matched layer (PML) absorbing boundary condition is employed in the X direction while periodic boundary conditions are utilized in the Y and Z directions. Every two boards are 2.1 mm away from each other in the Z direction. The dimensions of each grid in the FDTD simulation are 0.15 mmx y za a a . Besides, the

inductors and capacitors used as heterogeneous elements are lumped elements and the size of each is less than one grid in the FDTD simulation.

3.1 LHM doped with inductors (LDLHM)

LDLHM is designed by setting continuous inductors along the propagation direction and these inductors are on the surface where SRRs are located. The value of each inductor is

111 10 HL . However, the other side of the board only has wires just as shown in Fig. 3(b). Fig. 4(a) gives a close look at this LDLHM. Three rows of continuous inductors along the Y axis are embedded on the board surface with 21.6 mm long. The distance from the underside of the board to the edge of SRR is 0.6 mm and the distance between every two SRRs is 2.1 mm. The distance from the upper of the board to the edge of inductors is also 0.6 mm and the distance between two rows of inductors is 7.2 mm.

Fig. 4(b) shows the calculated transmission spectra of the traditional LHM, the inductor-only array, and the LDLHM respectively. After inductors inserted as heterogeneous elements, it is obvious that the transmission peak around 4.8 GHz is weakened and the pass-band of LHM from 8.0 GHz to 10.0 GHz is shifted to a frequency range from 8.7 GHz to 10.0 GHz. These changes can be explained by the following reasons. There is capacitance between every two columns of continuous inductors having performance of a high pass filter, which is validated by Fig. 4(b). Therefore, we

observe a stop band of the transmission spectrum starting from 0 Hz for the inductor-only array. In view of this point, the weakened resonance around 4.8 GHz is caused by the blocking effect of the introduced inductors. For the same reason, the bandwidth of LDLHM is narrowed also.

(a) Doped metamaterial with inductors in SRR

(b) Transmission spectra

Fig. 4 Scheme of LDLHM and the transmission spectra

On the other hand, the above phenomena may be explained by the shift of the electric plasma frequency. The electric plasma frequency fp of Pendry wire array is given as [3–4]

p

2 ln

cfaar

( 1 )

where a is the wire lattice spacing and r is the wire radius. It is well known that continuous wires can have a first forbidden band from 0 GHz to a plasma frequency. Coupled with the magnetic resonance of SRRs, a pass-band will appear under fp and the LHM’s resonance is occurred. Because the dielectric board is very thin, though continuous inductors are inserted on another side, the equivalent wire lattice spacing is decreased and the electric plasma frequency of LDLHM is increased according to Eq. (1). But the magnetic plasma frequency of SRR is not varied. With the increase of the electric plasma frequency, the overlap region of electric resonance and magnetic resonance becomes less. Consequently, the bandwidth of LDLHM is narrowed after introducing the inductors.

102 The Journal of China Universities of Posts and Telecommunications 2011

We then change the inductor value of LDLHM to investigate the resulted changes of transmission spectrum. By keeping other parameters with no change, we calculated the transmission spectra of three LDLHMs with the inductor value 11

1 1 10L H, 102 1 10L H, and 9

3 1 10L H

respectively. As shown in Fig. 5, with the value increasing of inductors, the pass-band is expanded, but there has little impact on the resonance at 4.8 GHz with 9

3 1 10L H.

According to the quasi-static model proposed by Maslovski [8], the electric plasma frequency for the wire array can also be given

p0 rave

12

fa L

( 2 )

where rave is the average relative permittivity of a unit cell,

and L represents the wires’ total inductance per unit length and is proportional to the magnetic flux . In the LDLHM, only the value of the inductors is varied and the equivalent lattice spacing is kept a constant. With the increase of the value of the inserted inductors, the magnetic flux in a unit cell will increase and the medium’s total inductance per unit length is increased. Ultimately, the electric plasma frequency of the LDLHM is decreased correspondingly. This decreased plasma frequency leads to the overlap region of electric resonance and magnetic resonance getting big, which gives rise to the expansion of the bandwidth.

Fig. 5 Transmission spectra of LDLHMs with vary of inductor values

When the inductors are located at the wires side shown in Fig. 6(a), the transmission spectra are obtained, which are given in Fig. 6(b) and similar to that of Fig. 5. And the inductor values are 11

1 1 10L H, 102 1 10L H, and

93 1 10L H respectively. It also demonstrated that the

bandwidth of LDLHM is expanded with the value increasing of inductors.

(a) Scheme of LDLHM

(b) Transmission spectra of LDLHM

Fig. 6 LDLHM and its transmission spectra

3.2 LHM doped with capacitors (CDLHM)

CDLHM is constructed by inserting one group of continuous capacitors between every two rows of wires. The SRRs side of the board is kept unchanged and with the configuration shown in Fig. 4. The value of each capacitor is C=1 pF. The distance from the underside of the board to the edge of wires is 1.5 mm and the distance from the underside of the board to the edge of capacitors is 3.3 mm. The front surface of CDLHM is shown in Fig. 7(a). The space between every two rows of capacitors is d1=3.6 mm and also the space between every two rows of wires is d2=3.6 mm. When EM waves are incident to CDLHM, the calculated transmission spectrum, also with that of traditional LHM and only capactior structures are shown in Fig. 7(b).

In contrast to the transmission spectrum of the traditional LHM, this CDLHM gives rise to a new pass-band from 10.5 GHz to 12.5 GHz. Moreover, the pass-band between 8.0 GHz and 10.0 GHz is shifted to the range from 7 GHz to 11.3 GHz. These phenomena can be explained as follows. The equivalent circuit model for a capacitor at microwave frequencies can be considered as a series of inductors and capacitors resonance elements (SLCREs). As there are inductor components in the introduced capacitor structure, the

Issue 3 LIU Wen, et al. / Design and analysis of quantized controllable doped left-handed materials 103

magnetic flux in a unit cell would increase and the total inductance per unit length of CDLHM is increased. As a result, the electric plasma frequency of the CDLHM is decreased, which gives rise to the expansion of the bandwidth. As mentioned earlier, continuous wires have performance of a high pass filter. But the only capacitor structures have features of low pass filter that shown in Fig. 7(b). So that the resonance around 4.8 GHz is not weakened. The new generated pass-band from 10.5 GHz to 12.5 GHz is caused by the interaction between the capacitors and the LHM structure.

(a) The front side of the CDLHM

(b) Transmission spectra of the LHM, capactior only and the CDLHM Fig. 7 Scheme of CDLHM and its transmission spectra

We then change the value of capacitor to investigate the quantized controllable CDLHM. By keeping other parameters of CDHLM invariant, we calculated the transmission spectra of three CDLHMs with the value of capacitors to be C1 =1 pF, C2 =2 pF, and C3 =3 pF. Fig. 8 illustrates the changes of the spectra. With the value increase of the inserted capacitors, there is little change for the resonance at 4.8 GHz. However, the pass-band is moved to low frequency and the bandwidth is narrowed. Besides, the new generated pass-band is also shifted to low frequency.

Because the thickness of the dielectric board is very thin, and the wires plane and SRRs plane are set very near, there are mutual inductance between the structures in the front and

back plane. According to Ref. [1], the equivalent circuit model of SRRs is depicted in Fig. 9. The left part is the equivalent circuit of SRRs, and the right part is the equivalent circuit of SLCREs. R, L and C are the equivalent resistance of the whole structure, equivalent inductance and equivalent capacitance of SRRs respectively, where L and C will determine the magnetic plasma frequency; Lg and Cg are the equivalent inductance and capacitance of SLCREs respectively; M is the mutual inductance.

Fig. 8 Transmission spectra of CDLHMs with different capacitor values

Fig. 9 The equivalent circuit model of SRR and SLCRE

The equivalent impedance of SRR circuit is deduced as [1]

1

1jj

Z R L ZC

UI

(3)

2 2g gwhere ( ) j 1 .Z M L C

Z is the additional capacitance or inductance. Obviously, if the values of Lg and Cg are chosen in a special rang to get

2g g 1L C , then Z will has the properties of a capacitor;

otherwise, it will has the properties of an inductance. Consequently, when the values of Lg and Cg are changed, the magnetic plasma frequency of SRRs will be changed correspondingly.

Because the magnetic plasma frequency is very great [1], and even if the values of Lg and Cg are small, Z also has the properties of a capacitor with

2 2g g( ) j 1/( )Z M L C , and the additional

capacitance 2 2g g1/( ) ( )C L C M is appeared.

Consequently, the magnetic plasma frequency of SRRs moves to a high frequency

1/ 2m 2 LCC C C . Apparently,

104 The Journal of China Universities of Posts and Telecommunications 2011

with the increase of Cg, the additional inductance C will become higher and m will decrease. As a result, the

frequency region, in which the real part of eff is negative, may shift to a lower frequency range and be narrowed at the same time.

When the capacitors are located at the SRRs side as shown in Fig. 10(a), the transmission spectra of this CDLHM are given in Fig. 10(b), which are similar to that of Fig. 8. And the capacitor values are 1 1 pFC , 2 2 pFC , and

3 3 pFC respectively. It also demonstrated that the

pass-band is moved to low frequency and the bandwidth is narrowed, with the value increasing of the inserted capacitors.

(a) Scheme of CDLHM

(b) Transmission spectra of CDLHM

Fig. 10 CDLHM and its transmission spectra

As mentioned in Ref. [9], the resonance frequency and bandwidth could be tuned by adjusting the structure size and its form. This method could offer considerable adjustment for the pass-band, however, it could not give a continuous adjusting in a wide range. In this paper, the inductors and capacitors are used as heterogeneous elements to achieve an adjustment method with quantized controllable for the pass-band of LHMs. The former is named as ‘rough adjustment’, and the latter is named as ‘fine adjustment’. It can be deduced from the above sections that the value of inductors and capacitors could affect the pass-band greatly, which gives rise to a continuous adjustment for the pass-band of LHMs, and the quantized controllable doped left-handed

materials are achieved. When these two methods are combined together, it will give a considerable and continuous adjusting method for the pass-band of LHHs.

4 Conclusions

In this paper, two sorts of DLHMs are proposed and analyzed. In contrast to the traditional LHM, the LDLHM shows that the resonance is weakened and the bandwidth is narrowed; whereas the CDLHM exhibits an expanded pass-band and gives rise to a new pass-band. Further, the transmission characteristics of the DLHMs with different value of inductors and capacitors have also been investigated. The simulation results show that the bandwidth is expanded with the increase of the value of inductors. When capacitors are introduced to LHM, the pass-band is shifted to low frequency and the bandwidth is narrowed with the increase of the value of capacitors. The transmission properties are good explanations to the concept of quantized controllable DLHMs.

In the future work, we will study the implementation of the DLHMs in microwave apparatus.

Acknowledgements

This work was supported by the National Natural Science Fundation of China (60871081), the Research Innovation Fund for College Students of Beijing University of Posts and Telecommunications (2010), the Beijing Natural Science Foundation (Design and Fabrication of Miniature Smart Antenna Based on Metamaterials, 4112039).

References

1. Zhang H X, Huang Y W, Zhao L, et al. Quantitatively controllable left-handed material doped with series LC resonance elements. Chinese Journal of Physics, 2010, 48(1): 103 116

2. Shelby R A, Smith D R, Schultz S. Experimental verification of a negative index of refraction. Science, 2001, 292 (5514): 77 79

3. Pendry J B, Holden A J, Robbins D J, et al. Low frequency plasmons in thin-wire structures. Journal of Physics Condensed Matter, 1998, 10 (22): 4785 4809

4. Pendry J B, Holden A J, Robbins D J, et al. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Transactions on Microwave Theory and Techniques, 1999, 47(11): 2075 2084

5. Smith D R, Kroll N. Negative refractive index in left-handed materials.Physical Review Letters, 2000, 85 (14): 2933 2936

6. Engheta N, Ziolkowski R W. A positive future for double-negative metamaterials. IEEE Transactions on Microwave Theory and Techniques, 2005, 53(4): 1535 1555

7. Bayindir M, Aydin K, Ozbay E, et al. Transmission properties of composite metamaterials in free space. Applied Physics Letters, 2002, 81 (1): 120 122

8. Zhang H X, Zhao L, Lu Y H, et al. Study on a sort of controllable nonlinear left-handed materials. Journal of Nonlinear Optical Physics & Materials, 2009, 18( 3): 441 456

9. Chevalier C T, Wilson J D. Frequency bandwidth optimization of left-handed metamaterial. NASA/TM— 2004-213403.2004

(Editor: WANG Xu-ying)