## Abstract

In this paper, we experimentally realize a one-dimensional RHM (Right-handed Material)-LHM (Left-handed Material) multi-frequency resonator that consists of a dual-negative-band LHM and air arranged in an X-band waveguide. Multi-resonant frequencies are observed within two left-handed bands of the LHM. The effects of the loss and the hyperbolic dispersion relation of LHM layer are discussed. The incorporation of such a LHM into the resonator design allows more flexibility to realize multi-resonance.

© 2006 Optical Society of America

## 1. Introduction

Since the observation of negative refraction in 2001 [1] using a prism configuration composed of artificial left-handed material (LHM), various applications of LHMs have been realized by experiments: the lens [2], the resonator [3], the coupler [4] and so on. As proposed in [5], for a one-dimensional (1D) RHM-LHM resonator composed of two parallel RHM and LHM layers (as shown in Fig. 1), the thickness of the resonator can be smaller than half wavelength. In [5], the author pointed out that for the structure shown in Fig. 1, the condition for resonance is:

where *n*
_{1} and *n*
_{2} are the refractive indexes, *μ*
_{1} and *μ*
_{2} are the permeabilities, and *t*
_{1} and *t*
_{2} are the thicknesses of the LHM and RHM layers, respectively, *k*
_{0} is the wave vector in free space.

In this paper, an LHM slab is put into a segment of X-band rectangular waveguide to build a resonator, in which the air layer acts as the RHM, shown in Fig. 2. In [3], the authors use 1D Ω -shaped LHM [6] as the LHM layer. The experimental results show that the resonator proposed in [3] has one resonant frequency in the negative band where both permittivity and permeability are negative. In the present paper, the dual-band S-shaped LHM reported in [7] is used as the LHM layer instead of the Ω-shaped sample. We show by experiment that there are multi-resonant frequencies within the two negative bands, and further discuss the resonance condition and the influence of the loss of the LHM sample.

## 2. Experimental Setup and Validations for LHM Sample

In Fig. 2, the LHM sample (10mm in *x* direction, 22.5mm in *y* direction and 4mm in *z* direction) is put into an standard X-band rectangular waveguide with a cross session of 22.86mm×10.16mm. The LHM sample contains 10 unit cells in the *y* direction, each with the same dimensions as reported in [7]. A short-circuit piston is connected to the right end of the rectangular waveguide. By pushing or pulling the piston along *z* direction, the length of the air layer can be controlled. A copper sheet with a small rectangular slot is connected to the the left end of the resonator, allowing power coupled into the resonator from the waveguide port. An Agilent 8722ES vector network analyzer is connected to the port to measure the *S*
_{11} parameter of the resonator.

Before performing the resonance experiment, the transmission property of the LHM sample placed in rectangular waveguide is measured (In [7], the experiment was conducted in a parallel plate waveguide). The measured curve is shown in Fig. 3, where two passbands (gray parts), from 9.6GHz to 11GHz and from 12GHz to 13.6GHz, are close to the negative passbands reported in [7], showing that the negative bands still exist in this configuration.

In [7], the dual-band S-shaped LHMwas used to make a prism with an angle of 18.4 degrees. The prism was placed in a beam-refraction experimental setup that was similar to that in [1], and the refractive indexes of the LHM sample were extracted from the far field power of the refracted beam shown in Fig. 3(b) of Ref. [7]. In this paper, we also do a similar simulation for comparison and further extract the refractive indexes, which are shown in Fig. 4, where the refractive indexes from the experiment and the corresponding simulation are both negative in two bands marked in grey, which agree with the two passbands obtained in the transmission experiment (Figure 3). The simulated and measured refractive index curves agree with each other, showing that the refractive indexes are negative within the two bands.

## 3. Experimental Resonance Results

For the dual-band S-shaped LHM sample arranged as Fig. 2, the permittivity and permeability are described as
$\left(\begin{array}{ccc}{\epsilon}_{1x}& 0& 0\\ 0& {\epsilon}_{1y}& 0\\ 0& 0& {\epsilon}_{1z}\end{array}\right)$
and
$\left(\begin{array}{ccc}{\mu}_{1x}& 0& 0\\ 0& {\mu}_{1y}& 0\\ 0& 0& {\mu}_{1z}\end{array}\right)$
correspondingly, where *ε*
_{1x} and *μ*
_{1y} are negative and the permeabilities along the other two directions are the same as that of air. Within the negative bands, for *TE*
_{10} mode inside the resonator, the resonance condition is modified as

where *μ*
_{2y} is the permeability of the RHM layer (air) in the *y* direction,
${k}_{1z}=-\sqrt{{\left({n}_{1}{k}_{0}\right)}^{2}-\left(\frac{{\mu}_{1y}}{{\mu}_{0}}\right){k}_{y}^{2}}$
and
${k}_{2z}=\sqrt{{\left({n}_{2}{k}_{0}\right)}^{2}-\left(\frac{{\mu}_{2y}}{{\mu}_{0}}\right){k}_{y}^{2}}$
are the *z* components of the wave vectors within LHM and RHM layers, respectively,
${n}_{1}=-\sqrt{{\epsilon}_{1x}{\mu}_{1y}}$
and *n*
_{2} is the refractive index of the air, *k _{y}*=(

*π*/

*a*) is the

*y*component of wave vector,

*μ*

_{0}is the permeability of air.

Firstly, a control experiment was conducted: the resonant frequency of a 11-millimeter-long resonator without the LHMsample, or totally filled with air (*d*
_{1}=0), is measured. The measurement shows that resonance occurs at 15.13GHz, showing that Eq. (2) is satisfied.

Secondly, keeping the total length 11mm unchanged, a 4-mm-thick LHM sample was inserted, leaving a 7-mm-thick air layer in the resonator. The experimental *S*
_{11} curve of the resonator from 9GHz to 13.5GHz is shown in Fig. 5, in which two resonant frequencies corresponding to the two negative bands, i.e., 10.1GHz and 12.84GHz are observed. When the wave propagates within the LHM layer at the two frequencies, the LHM behaves as a phase compensator because of the negative phase velocity along the z direction. Compared with the control experiment, the resonant frequencies are lowered. To achieve the same resonant frequencies, for the resonator only filled with air, the minimum total length of the resonator have to be 19.5mm for 10.1GHz and 13.6mm for 12.84GHz, which are both thicker than 11mm. Only with air, the two resonant frequencies 10.1GHz and 12.84GHz can’t exist at the same time.

Keeping the thickness of the LHM layer as 4mm, the resonant frequencies were measured for different thickness of the air layer by adjusting the piston, and the measurement result is shown in Table 1. The experimental data show that the RHM-LHM resonator has multiresonant frequencies within the two negative bands. Especially for the two resonant-frequency cases, one resonant frequency can be observed within each negative band.

From Fig. 4 and Table 1, the experimental refractive index of the dual-band S-shaped LHM and the resonant frequencies were obtained, from which *μ*
_{1y} can be estimated by applying Eq. (2) at each resonant frequency. The estimated and the retrieved *μ*
_{1y} from simulation data is shown in Fig. 6, where the retrieval algorithm proposed in [8] is adopted. In Fig. 6, the two curves match well with each other, indicating that the resonances are caused by the two negative bands of the LHM sample.

From Eq. (2), the resonant frequencies depend on *μ*
_{1y} and *n*
_{1}. The estimated *μ*
_{1y} shows that the LHM sample is highly dispersive within the two negative bands. In order to show the influence of the dispersion, the Drude model (
$\epsilon \left(\omega \right)=1-\frac{{\omega}_{\mathit{ep}}^{2}}{{\omega}^{2}+i{\delta}_{e}\omega}$
, *ω _{ep}*=14.7×10

^{9}×2

*π*rads/m) and Lorenz model ( $\mu \left(\omega \right)=1+\frac{0.17{\omega}_{m0}^{2}}{{\omega}_{m0}^{2}-{\omega}^{2}-i{\delta}_{m}\omega}$ ,

*ω*

_{m0}=11.884×10

^{9}×2

*π*rads/s) are used to match the retrieved permittivity and permeability curves of the second negative band of the LHM sample, respectively. The k surface of the LHM sample and air at two frequencies within the negative band is shown in Fig. 7. Since

*k*=(

_{y}*π*/

*a*) for two incident wave vectors

*k*

_{i1}and

*k*

_{i2}in the air at two frequencies respectively, by using phase matching, the magnitude of

*k*becomes larger when frequency is lowered, or in other words,

_{z}*μ*

_{1y}becomes more dispersive. Large

*k*

_{1z}indicates the possible existence of multi-resonant modes of the resonator, however, only one or two resonant frequencies are observed within each negative band in our experiment. This may be caused by loss: in [9], the authors point out that the quality of the RHM-LHM resonator decreases when the loss of LHM increases, further, the quality is mainly determined by the loss of permeability (even when the loss of permittivity is significantly larger than the one of the permeability) for the TE mode. The absolute values of the left side of Eq. (2) are calculated for three cases (

*δ*=1×10

_{m}^{3}×2

*π*rads/s,

*δ*=1×10

_{m}^{5}×2

*π*rads/s,

*δ*=1×10

_{m}^{6}×2

*π*rads/s, and

*δ*=1×10

_{e}^{7}×2

*π*rads/s), where the

*δ*and

_{m}*δ*represent electric and magnetic losses, respectively. The calculation results are shown in Fig. 8. If the absolute value of the left side of Eq. (2) is close to zero that corresponds to the dips in Fig. 8, the resonance occurs. In Fig. 8, the number of dips is reduced and the values of the dips deviate from zero when

_{e}*δ*increases, causing the decrease number of resonant modes.

_{m}For realizing multi-resonant frequencies in required bands, finding a RHM with appropriate dispersive properties is much more difficult than designing an LHM with specified negative bands. By changing the sizes of the rings of the S-shaped multi-band structure or scaling the total size of the structure, one can simply get the desired negative bands, which is more feasible for realizing multi-resonance.

## 4. Conclusion

In this paper, a multi-resonant RHM-LHM resonator based on a dual-band S-shaped LHM was demonstrated. The experimental results show that the resonant frequencies are within the negative frequency bands of the LHM sample. The refractive indexes extracted from the prism experiment and estimated *μ*
_{1y} from the experimental data show that the multi-resonant frequencies are caused by the left-handed properties of the LHM sample. The influence of the loss and the hyperbolic k-surface of the LHM sample were discussed, which show that they are important in the multi-resonant scheme. The flexibility of choosing the negative refraction bands for a S-shaped multi-band structure gives feasibility to realize multi-resonance in a required frequency range.

## References and links

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