## Multi-passband Tunneling Effect in Multilayered Epsilon-Near-Zero Metamaterials

Optics Express, Vol. 17, Issue 14, pp. 12183-12188 (2009)

http://dx.doi.org/10.1364/OE.17.012183

Acrobat PDF (349 KB)

### Abstract

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 OSA

## 1. Introduction

1. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70(4)**, 046608 (2004).
[CrossRef]

2. M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B **75(7)**, 075119 (2007).
[CrossRef]

3. M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. **97(15)**, 157403 (2006).
[CrossRef]

5. Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. **91(23)**, 234105 (2007).
[CrossRef]

6. R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. **100(2)**, 023903 (2008).
[CrossRef]

_{10}model [7

7. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. **100(3)**, 033903 (2008).
[CrossRef]

## 2. Simulations and result

_{10 }propagation constant (

*β*) of the metallic waveguide is described as follow Eq. (8):

*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.

8. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. **76(25)**, 4773 (1996).
[CrossRef]

9. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter **10(22)**, 4785–4809 (1998).
[CrossRef]

*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

10. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **71(3 3 Pt 2B)**, 036617 (2005).
[CrossRef]

11. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65(19)**, 195104 (2002).
[CrossRef]

*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.

*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 TE

_{10}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.

13. M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. **53(4)**, 1418–1430 (2005).
[CrossRef]

_{21}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.

## 3. Discussions

### 3.1 EZN tunneling effect and Fabry-Pérot resonance

*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

*f*, which is equal to

_{m}*mc*/(2

*n*

_{eff}*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

*n*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

_{eff}*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

_{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

*m*th 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

15. G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. **2(7)**, 251–252 (1966).
[CrossRef]

## 4. Conclusion

## Acknowledgements

## References and links

1. | R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

2. | M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B |

3. | M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. |

4. | M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B |

5. | Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. |

6. | R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. |

7. | B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. |

8. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. |

9. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter |

10. | D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

11. | D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

12. | M. David and Pozar, “transmission lines and waveguides,” in Microwave Engineering (John Wiley & Sons, New York, 2004). |

13. | M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. |

14. | J. Hupert, “Evanescent Mode Guide Filter and Tunnel-Effect Analogy,” IEEE Trans. Circ. Syst. |

15. | G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(160.0160) Materials : Materials

(230.7380) Optical devices : Waveguides, channeled

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 14, 2009

Revised Manuscript: June 29, 2009

Manuscript Accepted: June 29, 2009

Published: July 2, 2009

**Citation**

Liyuan Liu, Chenggang Hu, Zeyu Zhao, and Xiangang Luo, "Multi-passband tunneling effect in multilayered Epsilon-Near-Zero Metamaterials," Opt. Express **17**, 12183-12188 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12183

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### References

- R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046608 (2004). [CrossRef]
- M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007). [CrossRef]
- M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006). [CrossRef]
- M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76(24), 245109 (2007). [CrossRef]
- Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007). [CrossRef]
- R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008). [CrossRef]
- B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773 (1996). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998). [CrossRef]
- D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005). [CrossRef]
- D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). [CrossRef]
- M. David, Pozar, “transmission lines and waveguides,” in Microwave Engineering (John Wiley & Sons, New York, 2004).
- M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1418–1430 (2005). [CrossRef]
- J. Hupert, “Evanescent Mode Guide Filter and Tunnel-Effect Analogy,” IEEE Trans. Circ. Syst. 15, 279–280 (1968).
- G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. 2(7), 251–252 (1966). [CrossRef]

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