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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 26 — Dec. 21, 2009
  • pp: 24371–24376
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Transparency induced by coupled resonances in disordered metamaterials

Wei Tan, Yong Sun, Zhi-Guo Wang, Hong Chen, and Hai-Qing Lin  »View Author Affiliations


Optics Express, Vol. 17, Issue 26, pp. 24371-24376 (2009)
http://dx.doi.org/10.1364/OE.17.024371


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Abstract

We propose a scheme to induce transparency in one-dimensional disordered multilayers which are composed of negative permittivity and negative permeability metamaterials. First, analytical expressions for transparency condition are derived exactly, providing us a picture that complete tunneling in such kind of multiple-resonance system can be achieved if exponentially growing waves can compensate exponentially decaying waves. Second, a compensating method is used to realize this idea, and both simulations and experiments are performed in the microwave regime to confirm the theoretical analysis. Last, we have a discussion on how the coupling of resonances affects the transport properties of samples.

© 2009 OSA

1. Introduction

Propagation of waves through disordered systems is one of the most challenging topics in quantum and classical physics [1

1. J. B. Pendry, “Light finds a way through maze,” Physics 1, 20 (2008). [CrossRef]

]. In disordered medium, the destructive interference of scattering waves will give rises to Anderson localization, which has the characteristic that the system eigenfunctions decay exponentially (with a length scale, ξ) [2

2. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]

]. Surprisingly, for one-dimensional (1D) random systems, not all modes are localized. In 1987, Pendry [3

3. J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C Solid State Phys. 20(5), 733–742 (1987). [CrossRef]

] and Tartakovskii et al. [4

4. A. V. Tartakovskii, M. V. Fistul, M. E. Raikh, and I. M. Ruzin, “Hopping conductivity of metal-semiconductor-metal contacts,” Sov. Phys. Semicond. 21, 370–373 (1987).

] independently predicted the existence of delocalized modes that can extend over the sample via multiple resonances and have a transmission coefficient close to 1. These modes, called necklace states, have been addressed frequently through the past decades and experimentally confirmed in optical systems [5

5. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson localized 1D systems,” Phys. Rev. Lett. 94(11), 113903 (2005). [CrossRef] [PubMed]

,6

6. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. 97(24), 243904 (2006). [CrossRef] [PubMed]

].

Motivated by the idea of necklace states in conventional disordered systems, we propose a scheme to induce transparency in 1D random system made of two kinds of SNG metamaterials.

2. Theoretical model

Mj=(11Zj1Zj1)(exp(αjdj)00exp(αjdj))(11Zj1Zj1)1.
(2)

In the case of normal incidence, αj=|kj|=2πν|(εjμj)1/2|/c where kj is the wave vector in the jth layer and ν is the frequency of incident wave, Zj=(μj/εj)1/2 is the impedance, and dj (j=A,B) is the thickness of layer. At the resonance frequency ν0, the impedances satisfied ZA(ν0)=ZB(ν0), and the transmittance through an N-layer SNG disordered multilayer structure which is embedded in the substrate with impedance ZS can be calculated analytically,
T(ν0)=11+14[(ZS|ZA|+|ZB|ZS)sinh(NAαAdANBαBdB)]2,
(3)
where NA and NB are the total numbers of layer A and B, respectively, and NA+NB = N. It is clearly seen that 100% transmission at the resonance frequency ν0 can be achieved if and only if

NAαAdANBαBdB=0.
(4)

It is worth mentioning that reference [20

20. A. Aù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]

] painted a picture for tunneling of electromagnetic (EM) waves through an ENG-MNG pair that if the exponentially growing waves in one medium and the exponentially decaying waves in another get in balance, 100% transmission would appear. Here, we demonstrate that for multiple-resonance system, once the total exponentially growing waves completely compensate the total exponentially decaying waves [see Eq. (4)], the structure would be transparent. These results can be extended to a more complex disordered system where the thickness of each layer varying randomly as well as the position, if we substitute l=1NA(αAdA)l and l=1NB(αBdB)l for NAαAdA and NBαBdB, respectively.

In general, Eq. (4) is rarely satisfied because it is difficult for the exponentially growing waves to precisely balance the exponentially decaying waves in a random system. Hence, some of the energy will reflect back and the transmission coefficient cannot reach 1. In order to induce transparency in this disordered structure, we propose to place a compensating layer C in front of the sample to balance the growing and decaying waves. If the entire structure satisfies
NAαAdANBαBdB±αCdC=0.
(5)
where αC=|kC| with kC the wave vector of layer C and dC is the thickness, EM waves can completely pass through the entire structure (including the compensation layer and the disordered multilayer).

3. Simulations and experimental results

We consider a 24-layer system to exhibit our scheme in the microwave regime. The SNG metamaterials are realized by using transmission line structures [13

13. G. V. Eleftheriades, and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications, (John Wiley & Sons, Inc., New Jersey, 2005).

,14

14. C. Caloz, and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, (Wiley, New York, 2006).

]. The ENG layer is designed to have a length of dA=5.4mm and a loaded lumped inductor LA=10.0nH, while the MNG layer has dB=5.4mm and a loaded lumped capacitor CB=4.0pF. They are fixed on a FR-4 substrate with relative permittivity, 4.75, a thickness of 1.6 mm. The effective electromagnetic parameters of the SNG media can be described as [13

13. G. V. Eleftheriades, and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications, (John Wiley & Sons, Inc., New Jersey, 2005).

,14

14. C. Caloz, and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, (Wiley, New York, 2006).

]
εA=3.5713.08ν2,μA=1
(6a)
for the ENG layer, and
εB=3.57,μB=13.66ν2
(6b)
for the MNG layer, where ν is the frequency measured in GHz. Both ENG and MNG layers are effectively homogenous SNG materials in the frequency region between 0 and 1.9 GHz.

We take one sample for example, of which the sequence is ABA 2 BA 5 BA 4 B 4 A 2 B 2 A. The calculated transmission spectrum of this sample is shown in Fig. 1
Fig. 1 Calculated transmittance as a function of frequency for the sample before compensation (dash-dot line) and after compensation (solid line).
by a dash-dot line. Apparently, the transmittance is no higher than 15%. After we placed a compensating layer C in front of this sample, the transmittance through the entire structure reaches nearly 100% at the resonance frequency (solid line in Fig. 1). The compensating layer C selected for this sample is made of the same material as layer B and its thickness is dC=6dB.

In Figs. 2
Fig. 2 Intensity distributions inside the sample before compensation (a) and after compensation (b) corresponding to Fig. 1.
, we show the intensity distributions, I=|E|2, inside the sample before and after compensation, respectively. Before compensation, since the growing waves and the decaying waves of resonances are not in balance, the fields can hardly reach the right end of the sample, and incident wave easily escapes back. But after compensation, resonances are extended over the entire structure so that incident wave can pass through the sample by hopping from one resonance to another.

In experiments, losses in SNG metamaterials are inevitable. However, these losses lower the transmission coefficient but do not change the characteristics of the EM wave transport in our samples. In order to show clearly that whether most energy of incident wave enters the sample, we measure the reflectance R instead of the transmittance T, since resonances can also be exhibited in reflection [6

6. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. 97(24), 243904 (2006). [CrossRef] [PubMed]

]. The reflectance is simulated by the Agilent Advanced Design System (solid line in Fig. 3
Fig. 3 Numerical simulations (solid lines) and experimental results (open circles), showing resonance tunneling states in reflection spectra. (a) Reflectance of the sample before compensation. (b) Reflectance after compensation.
) and measured by Agilent 8722ES vector network analyzer (open circles in Fig. 3). It is clearly seen that before compensation [see Fig. 3(a)], most of the energy reflects back from the sample, whereas after the compensation [see Fig. 3(b)], there exhibit a dip in the reflection spectrum which denotes the appearance of resonance tunneling states. These results are in agreement with the theoretical predictions.

4. Discussions

We have proposed an approach to induce transparency in disordered SNG metamaterials by balancing multiple resonances at ENG-MNG interfaces. Now we discuss how the coupling of adjacent resonances affects the transport properties. Reference [19

19. P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77(11), 115332 (2008). [CrossRef]

] pointed out that the number of interfaces plays an important role for the wave propagation in SNG systems. In the case that layers A and B are periodically arranged, the coupling of the periodic resonances will lead to a wide band around ν0 if the matching conditions are satisfied [16

16. H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004). [CrossRef] [PubMed]

]. When randomness is introduced into the system, the total number of ENG-MNG interfaces is always less than that in a periodic sample containing same number of layers, and hence the coupling is weakened by disorder. From simple arguments, the transmission properties will vary from sample to sample, due to random number and positions of the interfaces.

In Figs. 4(a) and (c)
Fig. 4 Tunneling behaviors in different disordered samples. Top panels: calculated transmission spectrum (a) and field intensity distribution (b) for the first sample. Inset: transmission spectrum for periodic AB pairs. Bottom panels: transmission spectrum (c) and field intensity distribution (d) for the second sample.
we depict the transmission spectra of two 30-layer samples. Both of them satisfy Eq. (4): one has a sequence of B 2 A 3 B 2 ABABA 3 BA 2 B 3 AB 3 AB 2 A 3 and the other AB 5 A 2 B 2 AB 4 A 4 B 2 A 2 B 2 A 5. Figure 4(a) shows broadband transparency around ν0, while Fig. 4(c) shows high-Q transmission peaks. These result from the structure difference that the number of interfaces in the first sample is larger than that in the second one, and the resonances are more uniformly distributed. The corresponding field distributions are shown in Figs. 4(b) and (d). One can find that the one with high-Q transmission peaks has higher intensity modes. The spectrum for corresponding periodic AB pairs is shown as an inset in Fig. 4(a), which has the broadest transmission band in accordance with the analysis above.

In addition, we find that the resonance tunneling states in our study are analogous with necklace states in conventional disordered systems, for both of them show a picture of transmission through multiple resonances strung from one side of the system to the other. Note that conventional disordered systems are much more complicated so that the condition for transparency cannot be described by an equation as brief as Eq. (4). Nevertheless, the idea of compensating decaying waves would still be useful when applied to disordered systems containing multiple resonances, which needs further study.

5. Conclusion

In summary, we have designed a scheme to induce transparency in 1D disordered multilayers composed of two different SNG metamaterials. A compensating layer is placed in front of the disordered sample in order to balance exponentially growing waves and exponentially decaying waves of multiple resonances, and thus achieve complete tunneling. Experiments were performed in the microwave regime, which are in agreement with theoretical predictions. Moreover, we discussed how the coupling of resonances affects the transmission properties.

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grants No. 10874129 and No. 10634050, National Basic Research Program (973) of China under Grant No. 2006CB921701, and Shanghai Science and Technology Committee under Grants No. 08dj1400300 and No. 07dz22302.

References and links

1.

J. B. Pendry, “Light finds a way through maze,” Physics 1, 20 (2008). [CrossRef]

2.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]

3.

J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C Solid State Phys. 20(5), 733–742 (1987). [CrossRef]

4.

A. V. Tartakovskii, M. V. Fistul, M. E. Raikh, and I. M. Ruzin, “Hopping conductivity of metal-semiconductor-metal contacts,” Sov. Phys. Semicond. 21, 370–373 (1987).

5.

J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson localized 1D systems,” Phys. Rev. Lett. 94(11), 113903 (2005). [CrossRef] [PubMed]

6.

K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. 97(24), 243904 (2006). [CrossRef] [PubMed]

7.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

8.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

9.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

10.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

11.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]

12.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

13.

G. V. Eleftheriades, and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications, (John Wiley & Sons, Inc., New Jersey, 2005).

14.

C. Caloz, and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, (Wiley, New York, 2006).

15.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003). [CrossRef] [PubMed]

16.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004). [CrossRef] [PubMed]

17.

Y. Weng, Z. G. Wang, and H. Chen, “Band structures of one-dimensional subwavelength photonic crystals containing metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046601 (2007). [CrossRef] [PubMed]

18.

A. A. Asatryan, L. C. Botten, M. A. Byrne, V. D. Freilikher, S. A. Gredeskul, I. V. Shadrivov, R. C. McPhedran, and Y. S. Kivshar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99(19), 193902 (2007). [CrossRef] [PubMed]

19.

P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77(11), 115332 (2008). [CrossRef]

20.

A. Aù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]

21.

P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67(4), 423–438 (1977). [CrossRef]

OCIS Codes
(260.2160) Physical optics : Energy transfer
(260.5740) Physical optics : Resonance
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: October 28, 2009
Revised Manuscript: December 3, 2009
Manuscript Accepted: December 3, 2009
Published: December 18, 2009

Citation
Wei Tan, Yong Sun, Zhi-Guo Wang, Hong Chen, and Hai-Qing Lin, "Transparency induced by coupled resonances
in disordered metamaterials," Opt. Express 17, 24371-24376 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24371


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References

  1. J. B. Pendry, “Light finds a way through maze,” Physics 1, 20 (2008). [CrossRef]
  2. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]
  3. J. B. Pendry, “Quasi-extended electron states in strongly disordered systems,” J. Phys. C Solid State Phys. 20(5), 733–742 (1987). [CrossRef]
  4. A. V. Tartakovskii, M. V. Fistul, M. E. Raikh, and I. M. Ruzin, “Hopping conductivity of metal-semiconductor-metal contacts,” Sov. Phys. Semicond. 21, 370–373 (1987).
  5. J. Bertolotti, S. Gottardo, D. S. Wiersma, M. Ghulinyan, and L. Pavesi, “Optical necklace states in Anderson localized 1D systems,” Phys. Rev. Lett. 94(11), 113903 (2005). [CrossRef] [PubMed]
  6. K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, A. Z. Genack, B. Hu, and P. Sebbah, “Localized modes in open one-dimensional dissipative random systems,” Phys. Rev. Lett. 97(24), 243904 (2006). [CrossRef] [PubMed]
  7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  8. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]
  10. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
  11. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef] [PubMed]
  12. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
  13. G. V. Eleftheriades, and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications, (John Wiley & Sons, Inc., New Jersey, 2005).
  14. C. Caloz, and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, (Wiley, New York, 2006).
  15. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90(8), 083901 (2003). [CrossRef] [PubMed]
  16. H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6), 066607 (2004). [CrossRef] [PubMed]
  17. Y. Weng, Z. G. Wang, and H. Chen, “Band structures of one-dimensional subwavelength photonic crystals containing metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046601 (2007). [CrossRef] [PubMed]
  18. A. A. Asatryan, L. C. Botten, M. A. Byrne, V. D. Freilikher, S. A. Gredeskul, I. V. Shadrivov, R. C. McPhedran, and Y. S. Kivshar, “Suppression of Anderson localization in disordered metamaterials,” Phys. Rev. Lett. 99(19), 193902 (2007). [CrossRef] [PubMed]
  19. P. Han, C. T. Chan, and Z. Q. Zhang, “Wave localization in one-dimensional random structures composed of single-negative metamaterials,” Phys. Rev. B 77(11), 115332 (2008). [CrossRef]
  20. A. Aù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antenn. Propag. 51(10), 2558–2571 (2003). [CrossRef]
  21. P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67(4), 423–438 (1977). [CrossRef]

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