# A Novel Dual-Band Left-Handed Metamaterial Design Method

Authors: Si Li; Wenhua Yu; Atef Z. Elsherbeni; Wenxing Li; Yunlong Mao
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

A novel dual-band left-handed metamaterial (LHM) design method is proposed in this paper. Unlike other methods, where the designers focused their attentions on designing single LHM unit with multiple electric/magnetic resonances or combining multiple different LHM units together, the proposed method in this paper introduces an additional magnetic resonance to extract negative permeability, taking advantage of the areas between neighboring units. In this paper, we first designed a single-band single negative metamaterial (μ<0) and then connected neighboring units with metallic wires. This connection introduces a magnetic resonance that extracts another frequency band with negative permeability. With the help of arrayed metallic wires printed on the other side of the substrate, we successfully get a dual-band LHM. The proposed structures are analyzed with equivalent circuits and verified with simulations.


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## Body

## 1. Introduction

Metamaterial (MTM) is a kind of engineering materials which exhibits exotic physical properties that are not encountered in nature. MTMs which have negative permittivity and permeability together are also referred to as left-hand metamaterials (LHMs). LHMs have a series of fantastic characteristics such as negative refractive index and backward waves, which draw great attention all over the world. MTMs have been successfully employed in many areas such as cloaking, absorbing materials [1, 2], and antenna designs [3, 4].There are 2 kinds of realizations of metal-based LHMs in the microwave regime. The first kind of metal-based LHMs is implemented with magnetic resonances and arrayed metallic wires. The magnetic resonances, typically the split ring resonators (SRRs), are able to extract negative permeability at a specified frequency band, while the arrayed metallic wires are capable of providing negative permittivity at frequencies lower than their plasma frequency [5]. The other kind of LHMs is realized with the combination of electric LC (ELC) resonators [6, 7], which generate negative permittivity, and magnetic resonator. A common character of both LHMs is that the negative permeability is extracted from magnetic resonators.Most of the proposed LHMs only have one left-handed (LH) passband. In contrast, there are only a few studies on multiband (MB) LHMs. Chen et al. [8] printed an extended “S shaped” metallic pattern on both sides of an FR4 substrate, which exhibits 2 LH bands. However, for the purpose of containing multiple electric and magnetic resonances in a single unit, the cell size is relatively large. Wang et al. [9] stacked 2 layers of single-band (SB) LHM units with different geometrical dimensions to realize dual-band LH property but resulting in the reduction of transmission. Ekmekci et al. [10] stacked 3 layers of SRRs with different number of splits to extract two single negative (μ<0) bands but the transmission character in this work is also not good. Turkmen et al. [11] proposed a nested “U-shape” resonator that exhibits two or three LH bands according to the number of “U-shapes.” However, neither the electric nor the magnetic resonances are strong enough that the LH bandwidth is narrow and the LH transmission character is bad. Xu et al. [12] proposed a fractal “tree-shape” pattern featuring 3 magnetic resonances and an electric resonance apart from the lower metal plasma response. This complex structure is verified to have 2 LH bands with measurements. Liu et al. [13] demonstrated a complex nonlinear dual-band LHM which is geometrically similar to that proposed in [8]. Zhou et al. [14] proposed a double “Z-shaped” structure which exhibits 2 LH passbands under parallel incidence and 3 LH passbands under normal incidence.It can be concluded from the above-mentioned works that the main challenge facing MB LHMs is how to excite multiple magnetic resonances. In this work, we propose a novel method to design a DB LHM. First of all, we designed a single-band (SB) MTM with onlyμ<0. Then we tactfully took advantage of the area between neighboring MTM units to introduce an additional magnetic resonance. Thereby, we get a DB MTM, which can be easily turned into a DB LHM by printing metallic wires on the other side of the substrate [15]. The proposed structures are analyzed with equivalent circuits and verified with simulations. This method could be important guidance in the design of MB LHMs.


## 2. Single Negative Metamaterial

First of all, we designed a single-band single negative metamaterial (SNG) only with negative permeability. The geometry of the proposed structure is displayed in Figure1, where we have a=5 mm, b=1.5 mm, c=1.85 mm,d =3 mm, g1 = 0.1 mm, g2 = 0.1 mm, andl = 1 mm. The width of the lines is w = 0.2 mm, and the thickness ist = 0.018 mm. This pattern is printed on a 0.4 mm thick square FR4 substrate which has ɛr = 4.2.Figure 1
Geometry illustration of the proposed dual-band SNG.


### 2.1. Equivalent Circuit Analysis

The equivalent circuit for this structure is displayed in Figure2.Figure 2
Equivalent circuit of the structure in Figure1.According to strip line theory [16], the inductance of a strip line can be calculated using(1)L=2×10-4llnlw+t+1.193+0.2235w+tl,where l is the length of the strip line, w is the width, and t is the thickness. Assume that we have a function, “Cal_induc(w,l,t),” to represent (1); then the inductance can be calculated as(2)L1+L2=Cal_inducw,c+d2-w-g12+l,t,which is 3.4 nH. The total inductance for this circuit is Lm1=4(L1+L2)=13.6 nH.The capacitance per unit length of paralleled strip lines is calculated using [16, 17](3)Cpul=ε0εeFk,where ε0 is the permittivity of free space; εe and F(k) are calculated as(4)εe=1+εr-1Fk2Fk1,Fk=1πln⁡21+k′1-k′,0<k≤12,πln⁡21+k1-k-1,12<k≤1,where k=a/b, a=g2/2, b=a+w3, k′=1-k2, and k1=sinh⁡πa/2h/sinh⁡(πb/2h). Assume that there is a function, “Cal_cap(w,g,h,ɛr),” to represent (3), where w is the width and g is the gap of the paralleled wires, h is the thickness of the substrate, and ɛr is the relative permeability of the substrate; then we will have(5)C1=b×Cal_capw,g2,h,εr,C2=l×Cal_capw,g1,h,εr.It can be calculated that C1=0.06 pf and C2=0.04 pf. Hence, the total capacitance is(6)Cm1=C1C22C1+C2.The resonant frequency is calculated using(7)fm1=12πLm1Cm2,which is 12.1 GHz.


### 2.2. Verification with Simulations

To verify the single negative property of this structure, we ran a full wave simulation using HFSS version 15. The simulation domain and the incidence of a plane wave are illustrated in Figure1. The polarization of the electric field and magnetic field is in the vertical and horizontal directions, respectively. The light green box is an air box. The top and bottom boundaries are defined as PEC, while the front and back boundaries are defined as PMC. The simulatedS parameters are displayed in Figure 3.Figure 3
The simulatedS parameters.As is known to all, SNGs are not capable of power transmission. In Figure3, S21 reaches a peak at 12.8 GHz, indicating that single negative property may happen there. The effective parameters retrieved using “S parameter retrieval” method [18, 19] are displayed in Figure 4. Negative permeability can be observed at a frequency range from 12.18 GHz to 13.37 GHz, while the permittivity stays positive in the discussed frequency spectrum. The magnetic resonant frequency is 12.8 GHz, which is a little higher than our numerical analysis. The absolute single negative bandwidth is 1.19 GHz and its relative bandwidth is 9.3%.Figure 4
Retrieved effective (a) permittivity “ɛ,” (b) permeability “μ,” (c) impedance “Z,” and (d) refractive index “n.” The black solid lines refer to the real part of these parameters, while the red dashed lines refer to the imaginary part.
(a)
(b)
(c)
(d)


## 2.1. Equivalent Circuit Analysis

The equivalent circuit for this structure is displayed in Figure2.Figure 2
Equivalent circuit of the structure in Figure1.According to strip line theory [16], the inductance of a strip line can be calculated using(1)L=2×10-4llnlw+t+1.193+0.2235w+tl,where l is the length of the strip line, w is the width, and t is the thickness. Assume that we have a function, “Cal_induc(w,l,t),” to represent (1); then the inductance can be calculated as(2)L1+L2=Cal_inducw,c+d2-w-g12+l,t,which is 3.4 nH. The total inductance for this circuit is Lm1=4(L1+L2)=13.6 nH.The capacitance per unit length of paralleled strip lines is calculated using [16, 17](3)Cpul=ε0εeFk,where ε0 is the permittivity of free space; εe and F(k) are calculated as(4)εe=1+εr-1Fk2Fk1,Fk=1πln⁡21+k′1-k′,0<k≤12,πln⁡21+k1-k-1,12<k≤1,where k=a/b, a=g2/2, b=a+w3, k′=1-k2, and k1=sinh⁡πa/2h/sinh⁡(πb/2h). Assume that there is a function, “Cal_cap(w,g,h,ɛr),” to represent (3), where w is the width and g is the gap of the paralleled wires, h is the thickness of the substrate, and ɛr is the relative permeability of the substrate; then we will have(5)C1=b×Cal_capw,g2,h,εr,C2=l×Cal_capw,g1,h,εr.It can be calculated that C1=0.06 pf and C2=0.04 pf. Hence, the total capacitance is(6)Cm1=C1C22C1+C2.The resonant frequency is calculated using(7)fm1=12πLm1Cm2,which is 12.1 GHz.


## 2.2. Verification with Simulations

To verify the single negative property of this structure, we ran a full wave simulation using HFSS version 15. The simulation domain and the incidence of a plane wave are illustrated in Figure1. The polarization of the electric field and magnetic field is in the vertical and horizontal directions, respectively. The light green box is an air box. The top and bottom boundaries are defined as PEC, while the front and back boundaries are defined as PMC. The simulatedS parameters are displayed in Figure 3.Figure 3
The simulatedS parameters.As is known to all, SNGs are not capable of power transmission. In Figure3, S21 reaches a peak at 12.8 GHz, indicating that single negative property may happen there. The effective parameters retrieved using “S parameter retrieval” method [18, 19] are displayed in Figure 4. Negative permeability can be observed at a frequency range from 12.18 GHz to 13.37 GHz, while the permittivity stays positive in the discussed frequency spectrum. The magnetic resonant frequency is 12.8 GHz, which is a little higher than our numerical analysis. The absolute single negative bandwidth is 1.19 GHz and its relative bandwidth is 9.3%.Figure 4
Retrieved effective (a) permittivity “ɛ,” (b) permeability “μ,” (c) impedance “Z,” and (d) refractive index “n.” The black solid lines refer to the real part of these parameters, while the red dashed lines refer to the imaginary part.
(a)
(b)
(c)
(d)


## 3. Dual-Band LHM Implementation

In the numerical analysis of split ring resonators (SRRs) [20], the effective permeability is closely related to a coefficient, “F,” which is the fractional area of a unit cell occupied by the interior of the split ring. The larger “F” is, the stronger the magnetic resonance will be. Therefore, in most designs, “F” is set up very close to 1, which overrode the fact that the area between neighboring units can also be taken advantage of to build additional magnetic resonance. Particularly, when this area is large enough, the corresponding magnetic resonance should also be strong enough to extract negative permeability. Hence, following the design principle in [21] which claimed that the equivalent circuit for a magnetic resonator should be mirror-symmetry single-loop, we introduced an additional magnetic resonance easily by connecting the neighboring SNG units with metallic wires. The updated structure is displayed in Figure 5, where we have two neighboring units in the “E” direction connected. The pattern beyond the blue dashed line, which is referred to as “R1,” is the original magnetic resonant unit, while the pattern beyond the red dashed line, which is referred to as “R2,” is the newly built magnetic resonant unit.Figure 5
Metallic structure of the dual-band MTM.


### 3.1. Equivalent Circuit Analysis

In the previous discussion, we already analyzed the equivalent circuit of the original magnetic resonator, “R1.” Without the consideration of interactions between “R1” and “R2,” we can also analyze “R2” with equivalent circuit. The equivalent circuit is displayed in Figure6.Figure 6
Equivalent circuit for “R2.”Following (1), the inductance can be calculated as(8)2L1+L3=Cal_inducw,2c+a-d,t,which is 5.1 nH. The total inductance for this circuit is Lm2=2(2L1+L3)=10.2 nH, while the total capacitance is Cm2=C1/2=0.032 pF. Hence, the resonant frequency is 8.84 GHz.This connection also brings an electric resonance. Since the domain size is much smaller than the wavelength of the incidence, the external electric field across a unit can be approximated asU=Eza=RI+(-jωL)I+I/(-jωC), where Ez is the external electric field per length,Iis the induced current,R and L are the total parasitic resistance and inductance, respectively, and C is the total capacitance by the gap. Then, the volume current density Jv in each unit can be homogenized as(9)Jv=Ia2=jω/aLω2-1/LC+jωR/LEz=σ′Ez.Meanwhile [22](10)εeff=1+jσ′ωε0.Hence,(11)εeff=1-ω2-1/LC/aLε0-jωR/aL2ε0ω2-1/LC2+ωR/L2.From (11), at the resonant frequency, the real part of the effective permittivity is 1. With the increase of the frequency, the real part of the effective permittivity is getting smaller and it may be negative at frequencies higher than the resonant frequency. The equivalent circuit for this electric resonator is illustrated in Figure 7.Figure 7
Equivalent circuit of electric resonator.The inductance can be approximated as(12)L4=Cal_inducw,a2-g12-w+l,t,which is 2.54 nH. Hence, the electric resonant frequency is 10.8 GHz.By now, both the magnetic resonances and the electric resonator have been independently analyzed with equivalent circuits. From the numerical analysis, we can predict that this structure exhibits single negative property at frequencies around 8 GHz and double negative property at frequencies around 12 GHz. Further verifications are still needed through simulations.


### 3.2. Verification with Simulations

The simulation is also carried out with HFSS. The set-up for the simulation is exactly the same as before. The simulatedS parameters are displayed in Figure 8, and the corresponding retrieved effective parameters are displayed in Figures 9(a)–9(d).Figure 8
SimulatedS parameters of the proposed structure.Figure 9
Retrieved effective (a) permittivity “ɛ,” (b) permeability “μ,” (c) impedance “Z,” and (d) refractive index “n.” The black solid lines refer to the real part of these parameters, while the red dashed lines refer to the imaginary part.
(a)
(b)
(c)
(d)In Figure8, there are 2 passbands where S11 is smaller than −10 dB, and the peaks happen at around 7 GHz and 12 GHz, respectively. A strong decrease of S21 can be observed at frequencies around 8 GHz, which identifies that power can hardly transmit at these frequencies.It can be observed from Figure9 that the real part of permittivity is negative at frequencies higher than 11.33 GHz in the discussed frequency range. On the other hand, the real part of permeability is negative at two frequency spectrums, from 7.33 GHz to 8.57 GHz and from 11.58 GHz to 13.68 GHz. Therefore, this structure exhibits single negative property at frequencies from 7.33 GHz to 8.57 GHz and a LH property at frequencies from 11.58 GHz to 13.68 GHz.This DB MTM can be easily turned into a DB LHM if we print arrayed metallic wires with higher plasma frequency than 8.57 GHz on the other side of the substrate [14].


### 3.3. Dual-Band LHM Implementation

An optimized structure of the metallic wires printed on the other side of the substrate is displayed in Figure10, where e = 3.4 mm andf= 0.4 mm.Figure 10
Geometry of the metallic wires printed on the other side of the substrate.The simulatedS parameters for the updated structure are displayed in Figure 11, and the corresponding effective parameters are displayed in Figure 12. Double negative properties are observed at frequency ranges approximately from 7.87 GHz to 8.60 GHz and from 12.74 GHz to 14 GHz. The corresponding relative bandwidths are 8.8% and 9.4%, respectively.Figure 11
SimulatedS parameters of the DB LHM.Figure 12
Retrieved effective (a) permittivity “ɛ,” (b) permeability “μ,” (c) impedance “Z,” and (d) refractive index “n.” The black solid lines refer to the real part of these parameters, while the red dashed lines refer to the imaginary part.
(a)
(b)
(c)
(d)The mutual coupling between the metallic structures on both sides of the substrate decreases the equivalent inductance, causing both the magnetic resonances and the electric resonance to shift to higher frequencies and also the reduction of the MTM bandwidth.


## 3.1. Equivalent Circuit Analysis

In the previous discussion, we already analyzed the equivalent circuit of the original magnetic resonator, “R1.” Without the consideration of interactions between “R1” and “R2,” we can also analyze “R2” with equivalent circuit. The equivalent circuit is displayed in Figure6.Figure 6
Equivalent circuit for “R2.”Following (1), the inductance can be calculated as(8)2L1+L3=Cal_inducw,2c+a-d,t,which is 5.1 nH. The total inductance for this circuit is Lm2=2(2L1+L3)=10.2 nH, while the total capacitance is Cm2=C1/2=0.032 pF. Hence, the resonant frequency is 8.84 GHz.This connection also brings an electric resonance. Since the domain size is much smaller than the wavelength of the incidence, the external electric field across a unit can be approximated asU=Eza=RI+(-jωL)I+I/(-jωC), where Ez is the external electric field per length,Iis the induced current,R and L are the total parasitic resistance and inductance, respectively, and C is the total capacitance by the gap. Then, the volume current density Jv in each unit can be homogenized as(9)Jv=Ia2=jω/aLω2-1/LC+jωR/LEz=σ′Ez.Meanwhile [22](10)εeff=1+jσ′ωε0.Hence,(11)εeff=1-ω2-1/LC/aLε0-jωR/aL2ε0ω2-1/LC2+ωR/L2.From (11), at the resonant frequency, the real part of the effective permittivity is 1. With the increase of the frequency, the real part of the effective permittivity is getting smaller and it may be negative at frequencies higher than the resonant frequency. The equivalent circuit for this electric resonator is illustrated in Figure 7.Figure 7
Equivalent circuit of electric resonator.The inductance can be approximated as(12)L4=Cal_inducw,a2-g12-w+l,t,which is 2.54 nH. Hence, the electric resonant frequency is 10.8 GHz.By now, both the magnetic resonances and the electric resonator have been independently analyzed with equivalent circuits. From the numerical analysis, we can predict that this structure exhibits single negative property at frequencies around 8 GHz and double negative property at frequencies around 12 GHz. Further verifications are still needed through simulations.


## 3.2. Verification with Simulations

The simulation is also carried out with HFSS. The set-up for the simulation is exactly the same as before. The simulatedS parameters are displayed in Figure 8, and the corresponding retrieved effective parameters are displayed in Figures 9(a)–9(d).Figure 8
SimulatedS parameters of the proposed structure.Figure 9
Retrieved effective (a) permittivity “ɛ,” (b) permeability “μ,” (c) impedance “Z,” and (d) refractive index “n.” The black solid lines refer to the real part of these parameters, while the red dashed lines refer to the imaginary part.
(a)
(b)
(c)
(d)In Figure8, there are 2 passbands where S11 is smaller than −10 dB, and the peaks happen at around 7 GHz and 12 GHz, respectively. A strong decrease of S21 can be observed at frequencies around 8 GHz, which identifies that power can hardly transmit at these frequencies.It can be observed from Figure9 that the real part of permittivity is negative at frequencies higher than 11.33 GHz in the discussed frequency range. On the other hand, the real part of permeability is negative at two frequency spectrums, from 7.33 GHz to 8.57 GHz and from 11.58 GHz to 13.68 GHz. Therefore, this structure exhibits single negative property at frequencies from 7.33 GHz to 8.57 GHz and a LH property at frequencies from 11.58 GHz to 13.68 GHz.This DB MTM can be easily turned into a DB LHM if we print arrayed metallic wires with higher plasma frequency than 8.57 GHz on the other side of the substrate [14].


## 3.3. Dual-Band LHM Implementation

An optimized structure of the metallic wires printed on the other side of the substrate is displayed in Figure10, where e = 3.4 mm andf= 0.4 mm.Figure 10
Geometry of the metallic wires printed on the other side of the substrate.The simulatedS parameters for the updated structure are displayed in Figure 11, and the corresponding effective parameters are displayed in Figure 12. Double negative properties are observed at frequency ranges approximately from 7.87 GHz to 8.60 GHz and from 12.74 GHz to 14 GHz. The corresponding relative bandwidths are 8.8% and 9.4%, respectively.Figure 11
SimulatedS parameters of the DB LHM.Figure 12
Retrieved effective (a) permittivity “ɛ,” (b) permeability “μ,” (c) impedance “Z,” and (d) refractive index “n.” The black solid lines refer to the real part of these parameters, while the red dashed lines refer to the imaginary part.
(a)
(b)
(c)
(d)The mutual coupling between the metallic structures on both sides of the substrate decreases the equivalent inductance, causing both the magnetic resonances and the electric resonance to shift to higher frequencies and also the reduction of the MTM bandwidth.


## 4. Conclusion

In this paper, we introduced a novel method for the design of dual-band LHM. This design method is explained step by step from a single-band single negative metamaterial (μ<0) to a dual-band LHM. The key point for this method is the introduction of an additional magnetic resonance, which is skillfully operated taking advantage of the areas between neighboring units. Metallic structures that appeared in our design procedure are well analyzed with equivalent circuits and verified with simulations. This method has an important meaning in the design of multiband LHMs.



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