## Abstract

A theoretically, predicted bandgap corresponding to the zero volume-averaged refractive index is verified experimentally by measuring the scattering parameters of a one-dimensional layered stack composed of left-handed and right-handed materials. The unique property of the zero-*n̅* gap, also denoted zero order bandgap, is verified by experiments, showing that unlike a conventional photonic band gap, the frequency corresponding to the zero order gap remains unchanged when the periodicity is altered.

©2006 Optical Society of America

## 1. Introduction

Conventional photonic band gap (PBG) materials are a type of artificial composites with
periodically modulated dielectric function, and the photonic gaps are a consequence of Bragg scattering in these materials [1, 2]. More recently, a novel artificial metamaterial with simultaneously negative permittivity and negative permeability, termed left-handed medium (LHM), has been studied extensively. Various structures of such material have been realized by several research groups [3, 4, 5, 6, 7, 8, and 9]. A few works have been devoted to multilayered structures containing LHM. In one-dimensional periodic structure with alternative LHM and right-handed material (RHM), spurious modes with complex frequencies, discrete modes and photon tunneling modes are displayed in the band structure [10]. In Ref. [11], a more interesting phenomenon is predicted by analysis as well as by numerical simulation: when the volume average of the effective refractive index equals to zero, a new type of gap, denoted zero-*n̅* gap in Ref. [11], emerges. Different from conventional Bragg gap, which is inversely proportional to the lattice constant, such gap remains unchanged when the lengths of the LHM and the RHM increase or decrease simultaneously by the same factors.

In this letter, using a metamaterial sample which have been verified to possess a left-handed property over a wide frequency band, we performed a series of transmission experiments on various one-dimensional (1D) LHM-RHM layered stacks to verify the existence of the special bandgap in such structures and its unique property. Experimental results show that unlike conventional photonic band gap, the frequency corresponding to the zero order gaps remains unchanged when the periodicity is altered.

## 2. Theory

We first consider a one-dimensional (1D) periodic structure consisting of conventional PBG materials with a lattice constant *a* as shown in Fig. 1. In Fig. 1(a), each period of the layered structure consists of RHM1 with thickness *b*
_{1} and RHM2 with thickness *b*
_{2}. Assume that the refractive index of RHM1 and RHM2 are *n*
_{1} and *n*
_{2}, respectively, the average refractive index can be calculated as $\overline{n}=\frac{1}{a}{\int}_{0}^{a}n\left(x\right)\mathit{dx}=\frac{{n}_{1}{b}_{1}+{n}_{2}{b}_{2}}{a}$, where *a* = *b*
_{1}+*b*
_{2}. In general cases when there is a mismatch between the two RHM layers, when *k*
_{0}
*a* = (*n̅**ω* / *c*)*a* = *m*
*π* (Here m is an integer), we get the familiar Bragg condition. There are multiple values of *ω* that satisfy the Bragg condition, and at these frequencies band gaps occur. However, also as reported in Ref. [11], a more unusual gap appears when *n̅* = 0 in a multilayered stack consisting of alternative LHM and RHM (see fig 1 (b)), which becomes possible only after the advent of LHM. This special kind of gap distinguishes itself from the conventional PBG: the frequency at which zero-*n̅* gap occurs is independent of the lattice constant, while other Bragg frequencies scale with the lattice constant [11]. The gap can also be termed zero order band gap, since it satisfies the Bragg condition *k*
_{0}
*a* = (*n̅**ω*/ *c*)*a* = *mπ*, where *m*=0.

## 3. Experimental configuration

The double-S shaped metamaterial exhibits a left-handed property over a frequency range of about 10~16 GHz, the broadest LH passband known to date [8]. It is easy to find some gaps over such a wide band; therefore, it is an ideal candidate for the experimental verifications.

To constitute a one-dimensional (1D) LHM-RHM layered stack by incorporating double-S shaped structures, we fabricate metamaterial samples in the way shown in Fig. 2(a). In the sample, each metamaterial card is made by printing metallic patterns consisting of several unit cells on an FR4 substrate with a permittivity of *ε*_{r}
=4.6. The metallic patterns repeat themselves on the substrate, i.e., along the *x* direction with a periodicity of *a*, where *a* is larger than the total length *b*
_{1} of LHM. Therefore, there is a dielectric RHM part in each period with a length of *b*
_{2}. Such metamaterial card is then repeated in *y* direction, each one being compressed between the same size blank FR4 substrates. When the electromagnetic (EM) waves are fed along the *x* direction with a z-polarized electrical field, the slab sample behaves as a one-dimensional, three-period LHM-RHM layered stack. Other samples with different layers and periodicities used in the experiments are shown in Fig. 2(b), in which samples A and B are with an 11.2-mm-long LHM part, a 7.0-mm and an 8.4-mm-long dielectric RHM part in each period, respectively, while sample C and sample D are fabricated by doubling the lengths of LHM and RHM of sample A and B, namely, with a 22.4-mm-long LHM part, and a 14mm-long RHM part in the sample C, a 16.8mm-long RHM part in the sample D.

Two groups of experiments are carried out by measuring the transmission properties of the samples between aluminum plates in a similar manner to that reported in Ref. [8]. In the first group, the existence of the zero order bandgaps is verified by measuring the samples A and B. The second group of experiment is performed to further confirm the existence of zero order bandgaps by verifying their unusual property: while changing the lengths of LHM and RHM by the same factors (the lengths become the twice in the experiments), the zero order bandgaps remain unchanged. Samples C and D are used in this group of experiments.

## 4. Experimental results

The transmission properties (the solid lines) of the LHM-RHM layered samples in the first group of experiments are shown in Figs. 3(a), and 3(b), respectively. Over the frequency band of negative refraction (the dashed lines in the figures) of the double-S shaped LHM arise some gaps, whose measured power is below -45 dBm, some even reach as low as nearly -50 dBm. There are two gaps occurring at 11.75 GHz and 15.20 GHz in the measurement result of sample A, and also two gaps at 11.7 GHz and 15.15 GHz in the measurement result of sample B. To determine the type of each gap, we calculate *k*
_{0}
*a* / *π* and the volume average of refractive index *n̅* . In the calculation, we use the length *b*
_{1} as the effective length of the LHM in the layered LHM-RHM structure, and the length *b*
_{2} of the dielectric FR4 substrate as the effective length of the RHM. The effective refractive index *n*
_{1} of the LHM is extracted from the data obtained in the prism experiment in Ref.6, while the refractive index of the RHM is easily obtained from *n*
_{2} = √*ε*_{r}
=2.15, where *ε*_{r}
is the permittivity of the FR4 substrate. For each gap that occurs in the LHM-RHM layered structures, the average *n̅* and *k*
_{0}
*a* / *π* are summarized in Table 1. From the results, the gaps occurring at 15.20 GHz in the sample A and at 15.15 GHz in the sample B are zero order band gaps. The gaps at 11.75 GHz in the sample A and at 11.70 GHz in the sample B are-1 order gaps.

The transmission properties (solid lines) of the sample C and the sample D measured in the second group of the experiments are shown in Fig. 4(a) and Fig. 4(b), and the passbands of negative refraction of double-S shaped LHM are represented in the dashed lines. The gaps are at 15.0 GHz and 14.95 GHz, respectively, and compared with the gaps occurring at 15.20GHz and 15.15GHz which are supposed to be zero order gaps in the previous experiments, they both shift slightly to lower frequencies with 0.2 GHz. The gaps remain almost unchanged, while the supposed–1 order gaps in Fig. 3(a) and 3(b), all disappear. We thus further confirm the existence of zero order bandgaps in the periodic LHM-RHM structures, and verify the unique property.

In the above we considered the natural boundaries of the LHMs as the effective boundaries, and used the lengths of the LHMs as their effective lengths. Actually, due to the fringe effects in both ends of the LHMs, we can assume that an extra small length ∆ is added in either end to obtain a more reasonable effective length. In the first group of experiments, in which the LHM is comprised of four unit cells, ${b}_{1\mathit{\text{eff}}}^{\left(1\right)}$ =*b*
_{1}+2∆; while in the second group of experiments, the LHM is comprised of eight unit cells with a length of 2*b*
_{1}, therefore ${b}_{1\mathit{\text{eff}}}^{\left(2\right)}$ = 2*b*
_{1} + 2∆ . When the LHM and RHM lengths are increased simultaneously by a factor of two, 2${b}_{1\mathit{\text{eff}}}^{\left(1\right)}$ = 2*b*
_{1} + 4∆, and${b}_{1\mathit{\text{eff}}}^{\left(2\right)}$ < 2${b}_{1\mathit{\text{eff}}}^{\left(1\right)}$, while for the RHM, the effective length is twice of that in the first experiment ${b}_{2\mathit{\text{eff}}}^{\left(2\right)}$ = 2*b*
^{(1)}
_{2eff}. Considering the zero order bandgap condition *n̅* = 0 and that the effective refractive index *n*
_{2eff} of the RHM remains the same, ${n}_{1\mathit{\text{eff}}}^{\left(2\right)}$ of the LHM should be smaller than ${n}_{1\mathit{\text{eff}}}^{\left(1\right)}$. From the extracted *n*
_{1eff} curve of the double-S shaped LHM, the frequency corresponding to a smaller negative effective refractive index shifts to the left, so do the zero order bandgaps in the second group of experiments. Notice that in the above the subscripts “1” and “2” stand for the LHM and the RHM in each period of the LHM-RHM layered stacks, while the superscript “(1)” and “(2)” represent the first and second group of experiments, respectively.

## 5. Conclusions

In conclusion, we measured the scattering parameters of one-dimensional layered stack of the left-handed and right-handed materials, and observed zero order and -1-order gaps in the passband of the LH material. The theoretically predicted zero order bandgap as well as its unique property that the corresponding frequency is invariant upon a scale of length are verified in these experiments. Some other experiments related to LHM-RHM layered structures can be further carried out based on these experiments.

## Acknowledgments

This work is supported by Chinese Natural Science Foundation under contract 60371010, 60201001, 60531020 and 60277018.

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