1. Introduction

Dispersion describes the electromagnetic (EM) characteristics of a medium at different frequencies, and the phase velocity (PV) of an EM wave is the most important physical quantity characterizing the dispersion [1]. For an EM wave propagating in an isotropic medium is the speed of light in free space. Although PV is always lower than c in naturally occurring media that possess refractive indices greater than unity, we also know that it can be faster than c without violating any known physical laws, for example, in a plasma medium with a relative permittivity of $1-{\omega }_p^2/{\omega }^2$, where${\omega }_p$is known as the plasma frequency (PF). In an extreme and interesting case where the frequency of the EM wave overlaps with the PF, its PV will diverge due to a vanishing refractive index. An infinitely fast PV means that there is no phase delay along the wave propagation direction, which can be used in a number of technologically important applications, such as in-phase power divider, coherent EM wave sources, and highly directivity antennas. However, naturally occurring plasma media, for example ionized gases, are usually difficult to handle in laboratory and adapt to real applications. Fortunately, metamaterials provide a substitution of a man-made plasma medium that can be easily tuned and manipulated in laboratory.

Metamaterials are structured composites possessing unusual electromagnetic properties, such as permittivity and permeability not commonly seen in natural homogeneous materials. Many of these properties have enabled new applications and excited the imaginations of both physicists and engineers in the past years [2, 3]. In this paper, we focus on an anisotropic metamaterial, whose permeability along its optical axis exhibits a plasma-like frequency dependence, at a certain microwave frequency band. We show by theoretical analysis, parameter retrieval and experiment that when material loss is considered, while the PF is defined by the frequency where the real part of permeability approaches zero, ultra fast PV actually appears at a slightly lower frequency, due to the dispersion diagram change. Both parameter retrieval and experimental data show that within a narrow frequency band to the left of the PF, the inherent loss still keeps finite and is much less than that in the corresponding resonant region. In the real metamaterial sample, an ultra fast phase velocity of 24,440 times the speed of light in free space is measured, and negative phase propagation due to the only negative permeability is observed. The existence of such ultra fast phase velocity with finite loss perfectly explains how the highly directivity antennas based on near-zero refractive index metamaterial work [4–9], why the ε-near-zero material can be used for tunneling and squeezing electromagnetic energy through subwavelength narrow channels and waveguides [10], and can be further used in other applications such as in-phase wave divider [11] and coherent wave sources.

2. Theoretical analysis

We begin with a magnetically uniaxial effective medium with a scalar permittivity ${\epsilon }_r$ and a permeability tensor defined in an (x, y, z) Cartesian coordinates of

 

Fig. 1 Dispersion diagram of $k_x$in a plasma medium with (a) ${\gamma }_e=0$, ${\epsilon }_r=1$ and (b) ${\gamma }_e=0.002{\omega }_p$, ${\epsilon }_r=1+0.05\text{i}$, where$k_z=0$.

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3. Parameter retrieval

In the following simulation and experiment, we used a slab-shaped metamaterial sample consisting of multilayered printed circuit boards (PCBs) carrying unit cells made of metallic split-ring resonators (SRRs). Similar metamaterial slabs have been used in Ref [9]. to obtain highly directivity radiation around the PF. This kind of metallic pattern was first proposed in Ref [15]. for obtaining metamaterial showing effective negative permeability. In our realization illustrated in Fig. 2(a) , the metallic (copper) SRRs are printed on 1-mm-thick FR4 substrates, with a relative permittivity around 4.6 at microwave frequencies. Each board contains 82 unit cells along the x direction with a periodicity of 6 mm and 15 unit cells in the y direction with a periodicity of 10 mm. Eight layers of such PCBs are aligned along the z direction spaced by 16 mm to form a slab-shaped sample with a width of 500 mm along the x direction and a thickness of 105 mm in the z direction. In such a slab, for an incidence with z-polarized magnetic field, magnetic resonances can be induced at X-band microwave frequencies inside the metallic rings, yielding a complex dispersion with a negative band of permeability along the z direction. Meanwhile, there is no magnetic response in the x and y directions, allowing${\mu }_{rx}$ and ${\mu }_{ry}$to be regarded as unity. By utilizing the homogenization approach proposed in Ref [16, 17], the dispersion of ${\mu }_{rz}$can be retrieved from the scattering parameters obtained from a full-wave FDTD simulation for the real configuration of the slab, and the result is shown in Fig. 2(b). We see that when the frequency is between 9.3 GHz and 9.5 GHz (marked in gray), ${\mu }_{rz}$exhibits a dispersion very similar to a typical plasma medium with a PF at 9.43 GHz and a finite loss that is much smaller than that in the region centered around 9.2 GHz, which is the resonant frequency of the SRRs.

 

Fig. 2 (a) Photograph and the geometry of the SRRs consisting of the metamaterial sample. (b)The real and imaginary parts of ${\mu }_{rz}$ retrieved from simulation data.

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4. Experimental investigation

To experimentally verify the aforementioned analysis and conclusion, we measure the PV within a frequency band near the PF. In the measurement, a small coaxial connector (the inset in Fig. 3(a) ) with a protruded 15-mm-long internal conductor serving as a monopole antenna is placed in the center of the slab, and another identical connector is used as a receiving antenna. To measure the PV inside the metamaterial along the x direction (such that $k_z=0$), three points designated as a, b and c, which are equally spaced by 12 mm on the x-axis, are chosen. The phase at each point is measured using an Agilent 8722ES vector network analyzer, and the phase velocity along the x direction is calculated by $v_p=\omega /k=\omega \cdot \Delta d/\Delta \varphi$, where $\Delta d$ is the distance between two adjacent measurement points, i.e., 12 mm, and $\Delta \varphi$ is the corresponding phase difference.

 

Fig. 3 (a) Experimental setup. Eight horizontal parallel lines represent eight PCBs shown in Fig. 2(a). (b), (c) The measured phase and amplitude of the electric field at points a, b and c, separately.

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Figure 3(b) shows the phases measured ranging from 9.2 GHz to 9.6 GHz. We see that at the frequencies far away from a frequency around 9.32GHz, the phase difference between the points a, b and c increase. However, at the frequencies close to 9.32GHz, the phase differences become negligible and less frequency dependent. Particularly, at 9.3235 GHz, the phase difference almost vanishes, implying an existence of ultra fast PV. This 9.3235 GHz frequency is 0.9887 times of the retrieved PF at 9.43 GHz in Fig. 2(b), and thus fits quite well with previous analysis. It is also seen that in the adjacent narrow bands lower and higher than the 9.3235 GHz, the phase difference has different signs, which is also in accordance with the previous theoretical analysis, i.e., negative phase propagation could occur at the frequencies to the left of the 9.3235 GHz.

In addition to the phases, Fig. 3(c) shows the amplitude of the electrical field at the same measuring points. As expected, as the frequency is leaving 9.4 GHz (We can assume that the PF of the metamaterial is actually around this frequency) to higher frequencies, there behaves a good transmission due to a real and positive refractive index, while as the frequency is moving to the left, the amplitude is decaying due to the increasing imaginary part of the ${\mu }_{rz}$. The thing interesting is that in the region between 9.3 and 9.4 GHz, the decaying is actually very slow, and the amplitudes measured at points a, b and c are still very much higher than the sensitivity of the instrument, while in the region lower than 9.3 GHz, the decay is drastically accelerated. This trend well matches the variation of the imaginary part of ${\mu }_{rz}$ in Fig. 2(b) between 9.2 and 9.4 GHz, again implying that that in the small band containing the 9.3235 GHz frequency, the loss is finite and therefore the ultra fast PV phenomenon can be used in real, loss-insensitive applications.

In Fig. 4 , the PV curve calculated from the experimental data (in dashed red) is shown, where the y-axis represents $sign({\nu }_p)\times \lg (|{\nu }_p|/c)$. We see that at the frequency of 9.3235 GHz, a PV of 24,440c, where c is the speed of light in free space is experimentally obtained, and with the frequency moves to higher frequencies, the PV drastically falls. This variation trend of the PV versus frequency clearly reflects a plasma-like dispersion. As predicted before, when frequency moves from 9.3235 GHz toward lower frequencies, while the PV also drastically decreases, its direction is negative, meaning that in such lossy area, negative phase propagation, and therefore negative refraction, can be expected to occur without requiring simultaneously negative permittivity and permeability. Although the loss is substantial, the negative refraction at this frequency band is of interest and needs to be further investigated.

 

Fig. 4 Calculated PV curves based on the measured data (dashed red line) and the retrieved data in Fig. 2(b) (solid black line).

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For comparison, the PV calculated from the retrieved data is also shown in sold black. Under a sweeping interval of 0.01 MHz (versus the 0.1 MHz interval provided by Agilent 8722ES), the fastest PV of 4,300,000c appears at 9.3379 GHz, which is 0.014 GHz higher than the 9.3235 GHz in experimental case. Considering the unavoidable difference of the material and geometry parameters between the simulation and experiment, as well as the difference of 10 times the sampling intervals, the diversity is actually reasonable.

5. Conclusion

In short, we discussed the dispersion of a magnetically uniaxial metamaterial and measured experimentally a metamaterial sample consisting of periodically arranged SRRs exhibiting plasma-like dispersion in microwave frequencies. An ultra fast PV of 24,440c is observed and the PV can be negative in an adjacent band. In the mean time, the loss is inherent but is also finite. Experimental results fit quite well with theoretically analysis and parameter retrieval. The ultra fast phase velocity along the x direction can be used to perfectly explain how the highly directivity antennas based on such uniaxial metamaterial work, and can be used in other applications such as in-phase wave divider and coherent wave sources. Moreover, at optical frequencies, similar dispersive behaviors can be found in the band diagrams of photonic crystals, indicating that ultra fast PV can also be found in optics, and therefore can be used in optical applications.