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Recently, several experimental results verified the tunneling effect theory of that the electromagnetic energy can be squeezed through an ultra-narrow channel filled with epsilon-near-zero (ENZ) medium. However, the energy squeezing can be only achieved in a narrow region. Here, we present a full-wave simulation of the tunneling effect in multilayered channels full of thin ENZ metamaterials with different plasma frequencies. Thin metallic wires arrays with different radiuses are employed to form these effective ENZ media, whose plasma frequencies are different. The appearance of several passbands in the transmission curve verifies that multi-passband energy tunneling effect can be implemented by multilayer ENZ channels. There are two possible reasons for these peaks, one is the ENZ tunneling effect, and the other is the Fabry-Pérot resonance. For each transmission peak corresponding two-spatial maps of electric field are given, in order to distinguish the causes.
©2009 Optical Society of America
1. Introduction
In recent years, growing attention is paid to metamaterials with unusual permittivity and permeability due to their anomalous influence on wave propagation properties. Among them, a special class named epsilon-near-zero (ENZ) medium has some characteristic features, such as “static-like” character of the electromagnetic field and zero phase variation at various point in it, putting it in amazing application outlook like highly directive antenna, compact resonator, transparency and cloaking phenomena, and pattern reshaping and so on [1,2].
In 2006, Silveirinha and Engheta [3] put ENZ medium in the narrow channel to make electromagnetic wave squeeze through it on condition that one of these dimensions of the channel is electrically small, which achieved the tunneling effect. Moreover, they found that the ENZ medium is independent on the length and specific cross section geometry of the ENZ channel. Based on this effect, the perfect waveguide coupler can be realized. And then in Ref. 4, the detailed theory of super coupling has been investigated. Up to now, several research groups have experimentally demonstrated the tunneling effect in ENZ metamaterials. In these verifications, the complementary split ring resonators (CSRR) was adopted to construct effective ENZ medium, realizing the tunneling effect in the planar waveguide at certain frequency band [5,6], and Edwards et al utilized the effective zero permittivity of the hollow waveguide around the cutoff frequency of its TE10 model [7]. However, owing to the very narrow epsilon-near-zero bands of both metamaterials and hollow metallic waveguide, the electromagnetic energy just can be squeezed through a single narrow or unique passband of the “perfect coupler”.
Here we present a mimic demonstration that multi-passband of “perfect coupler” can be realized by adopting multilayered narrow ENZ channels, where wire medium is used to act as ENZ medium. Different tunneling channels are filled with ENZ metamaterials with different plasma frequencies. Additionally, we analyze the causes of several transmission peaks.
2. Simulations and result
The structure used to realize the multi-passband tunneling effect is shown in Fig. 1. In this structure, two rectangular metallic waveguides are connected by coordinate ultra narrow channels A and B, which are filled with ENZ materials, and space between the two channels is filled with metal bulk C. The TE10 propagation constant (β) of the metallic waveguide is described as follow Eq. (8):
where w is H-plane width, n is the relative refractive index of the dielectric filling the waveguide (in our simulation configuration the filling dielectric in waveguide is air), c is the speed of light in vacuum and f is the operating frequency. The size of waveguide is d=11mm along the E-direction and m=30mm along the k-direction. The thickness of metal walls of the waveguide is 1mm. The two channels filled with ENZ materials have the same dimension. The channel length along the propagation direction is p and separation is t=1mm much less than the spacing of the input and output waveguide. In order to realize the multi-passband, the ENZ metamaterials filled in different channels should have different plasma frequencies where the permittivity is nearly zero. In this case, the energy squeezing and super coupling in ultra-narrow spacing are expected at multiple frequency bands.
Fig. 1. (a) Schematic illustration of the simulation structure, consisting of four waveguide sections. Input and output waveguide connecting by two E-direction narrow waveguides filled with ENZ metamaterials. (b) Cross-sectional schematic of the setup. (c) The effective permittivity of the wire medium filling in narrow waveguides (inset: the unit cell of wire media). The ENZ media filling in the channel A is composed of Roges 5880 (ε=2.2) and arranged copper wires with radium r 1=0.3mm and plasma frequency f 0=8.68GHz, then the effective permittivity is shown by blue curve in (c). The red curve is for media r 2=0.5mm and f 0=10.06GHz, filling in the lower channel.
In this paper, thin-copper wires array with regular small lattice spacing is employed to act as the ENZ metamaterials, and it can be analyzed by the effective medium theory, according to which both element dimension and lattice spacing are much smaller than wavelength [8,9]. Owing to the modulation effect of thin wires on the electron density, the plasma frequency ω p can be depressed to the microwave region. In the upper and the lower channels, the radiuses of metallic wires are r 1=0.3mm and r 2=0.5mm respectively, and both lattice spacing are a=6.8mm. The spacing between the wires is filled with Rogers 5880 whose relative permittivity is 2.2. The unit cell of the wire media is shown by the inset illustration in Fig. 2. The CST microwave studio based on the Finite Integral Time Domain Method is employed to obtain the S-parameters for a single unit cell with periodic boundary in both the E-direction and the H-direction, and then the effective index n(ω) and wave impedance z(ω) are calculated by the S-parameters retrieval method [10,11]:
And the permittivity ε (ω) and permeability µ (ω) are related to n and z by the relations:
Then we obtain the plasma frequency where the permittivity ε (ω) equals to zero. The retrieved effective permittivity is shown in Fig. 1. The blue line is the real part for r 1, where the corresponding plasma frequency is 8.68 GHz, and the red is for r 2 whose plasma frequency is 10.05 GHz.
In the full-wave simulation, the tunneling system consists of the channel A filled withENZ medium r 1=0.3mm, and the channel B filled with ENZ medium r 2=0.5mm. Wires arrays in both channels are all connected with the ‘ceiling’ and ‘floor’ of the narrow wave guide, so these finite wires can be treated as infinitely long in the E-direction. In each narrow channels, there are 4 cells in the propagation direction and 14 cells in the H-direction, which makes the whole size of ENZ medium be p=4×a=27.2 mm and w=15×a=102 mm. In this case, the corresponding cutoff frequency of the fundamental mode is calculated to be 1.04 GHz [12]. The fundamental transverse-electric TE10 mode is considered here. Simulation of this multi-passband tunneling wave guide is performed with the CST microwave studio based on the Finite Integral Time Domain Method. The smallest mesh is set to be x×y×z=0.15mm×0.15mm×0.2mm for wires in both the channels. The simulated transmission coefficient S21 as a function of frequency for multi-channel is shown in Fig. 2 by the red curve. Further, the green (blue) curve shows the transmission spectrum in the tunneling system by closing channel B (A). Two apparent shifts (8.49GHz→9.89GHz and 10.5GHz→11.5GHz) are observed in the green and blue curves. These shifts are mainly attributed to the different plasma frequencies of the wire medium filled in two channels.
In Fig. 2, the red curve shows four amplitude peaks, which correspond to the transmission peaks of the single-layered. The transmission peaks of the multi-layered structure excellently agree with those of single-layered structure, indicating that the peaks of multi-layered structure are likely to come from combination of the two single-layered structures. It is clearly demonstrated that the multi-passband tunneling effect can be successfully implemented by multilayered ENZ metamaterials. Transmission peak 1 (8.49GNz) is slightly shifted to low frequency compared with the peak frequency 8.68GHz obtained from the simulation of unit cell for r1, and the same phenomenon is also seen at transmission peak 2. It can be explained by the following possible reasons: the finite number of unit cell the H-direction, the interface effects and the spatial dispersion [13].
For each transmission peak, the corresponding electric field amplitude results and the field distribution of electric vector extracted from the CST software are presented in Fig. 3. For more convenient discussion, peaks appearing in the transmission coefficient S21 for multilayered structure are numbered as insert in Fig. 2. The energy of peak 1 and peak 3 squeezes through channel A, as shown in Fig. 3(a), 3(b) and 3(d). The tunneling channel is B for peak 2 and peak 4, as indicated in Fig. 3(c) and 3(e). There results provide a further prove for the multi-passband tunneling effect based on the multilayered ENZ channels.
Fig. 2. Simulated transmission curves for multi-layered (red), r 1=0.3 single-layered (green) and r 2=0.5 single-layered (blue).
Fig. 3. (color online) Two-dimension mapper and the field distribution of electric vector results (a) and (f) Field distribution of tunneling effect for peak 1 at 8.49GHz. (b) and (g) Field distribution of Fabry-Pérot-like oscillation for peak 1 at 8.79GHz. (c) and (h) Field distribution for peak 2 at 9.89GHz. (d) and (i) Field distribution of Fabry-Pérot-like oscillation for peak 3 at 10.5GHz. (e) and (i) Field distribution of Fabry-Pérot-like oscillation for peak 4 at 11.5GHz.
3. Discussions
3.1 EZN tunneling effect and Fabry-Pérot resonance
It is worth pointing out that there are two reasons for these amplitude peaks: the ENZ tunneling effect and Fabry-Pérot resonance. The ENZ tunneling effect frequency is not sensibly shifted by a change of length of narrow channel along the k-direction, and possibly either by its geometry, while the Fabry-Pérot transmission resonances at higher frequency are sensitive to the above dimension variation. The resonance frequency of m th-order Fabry-Pérot cavity in the wired medium is assumed to be fm, which is equal to mc/(2neff p) (m is an integer, c is the light velocity in the vacuum, p is the size of narrow waveguide along the k-direction, and the neff is the effective refraction index of wired medium). It can be seen that the peak 1 is slightly divided into two sub-peaks. One peak at lower frequency is caused by the ENZ tunneling effect, and there is almost no phase difference at various points across the channel (shown as Fig. 3(f)). The other one at higher frequency results from the first-order Fabry-Pérot resonance, and its phase variety is about π across the narrow channel A (shown as Fig. 3(g)). However, the division occurring at peak 1 is not obvious in the peak 2, because of smaller difference of the ENZ tunneling frequency and the first-order Fabry-Pérot resonance frequency in the wired medium with r 2=0.5mm. This peak is a mergence of first-order Fabry-Pérot-like oscillation and the tunneling effect in channel B. So the peak 2 is a typical combination of a traveling wave and a standing wave. The phase here also varies π across the narrow channel B at this frequency, as shown in Fig. 3(h). For the same wired medium, the difference between the ENZ tunneling frequency and the first-order Fabry-Pérot resonance frequency can be changed by varying the size of the narrow channel along the k-direction. If the size is shorter, this difference will be larger. At higher frequencies, both the peak 3 at 10.5GHz and the peak 4 at 11.5GHz are caused the second-order Fabry-Pérot resonance, and the variation of phase with 2π can be observed across both channel A and B, as shown in Fig. 3 (i) and (j).
3.2 Bandwidth expanding and its restrictions
There are three passbands in the transmission spectrum of our structure. In the first and the third one the best coupling efficiencies are almost 99.5% at 8.495GHz and 99.6% at 11.4GHz, respectively. In the widest band, the maximum coupling efficiency is more than 99.5% at 9.92GHz. Most importantly, the band width (BW) is greatly expanded. For the single-layered case, the transmission band is from 9.57GHZ to 10.04GHz [See Fig. 2 BW|S21=0.5=∇f/f 0=4.8%; while for the multi-layered case, the enlarged pass band is as high as 1.11GHz (from 9.59GHz to 10.7GHz) [See Fig. 2 BW|S21=0.5=∇f/f 0=11%]. Although the combination of peak 2 and peak 3 expands the transmission band, this expansion is not always available because of the difference in physical mechanism between peak 2 and peak 3. Peak 2 comes from the tunneling effect of ENZ medium. They depend on the plasma frequency and the dimension of the narrow channels in E-direction rather than the size along the direction of propagation, which means that no matter the dimension p is increased or decreased, the ENZ tunneling transmission peak will not be shifted. Remarkably different from the tunneling effect transmission peak, peaks caused by the mth Fabry-Pérot-like oscillation will be shifted towards lower frequency with the expansion of the dimension p of ENZ metamaterials in propagation direction [14,15]. That is to say, the above expansion will disappear if the dimension p in propagation is different. So the dimension p should be appropriately designed for the enhancement of passband.
4. Conclusion
In conclusion, a method for squeezing energy in the multi-frequency bands is presented by adopting multi-layered ENZ medium. The simulation result presents the existence of several transmission peaks through different ENZ channels at the corresponding zero permittivity frequencies for different wire medium, which verifies the feasibility of this idea that multilayered ENZ metamaterials can be used to realize multi-passband. Moreover, it is expected that more layers ENZ medium with different plasma frequencies will lead to wider passbands, but the coupling between each channel cannot be neglected in this case. Even when the channels do not have the same geometry or are not regular, multi-passband tunneling effect is still feasible.
Acknowledgements
This work was supported in part by the Nation Basic Research Program (973) of China under Grant No. 2006CB302900, in part by the Chinese Nature Science Grant (No. 60727006 and No. 60678035).
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