## Losses in left-handed materials

Optics Express, Vol. 11, Issue 19, pp. 2397-2402 (2003)

http://dx.doi.org/10.1364/OE.11.002397

Acrobat PDF (84 KB)

### Abstract

Interest in negative refractive index, or left-handed (LH) materials, has escalated rapidly over the last few years and it now appears that useful LH materials may be realizable in the microwave region. However there is also considerable interest in LH materials for infrared and visible applications. The purpose of this paper is to explore the limitations of LH materials at short wavelengths due to inherent losses. Our conclusions are that it may be quite difficult to achieve useful LH materials at wavelengths less than about 10 microns using current approaches.

© 2003 Optical Society of America

## Introduction

1. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and µ,” Soviet Physics USPEKHI **10**509–514 (1968) [CrossRef]

2. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. **76**, 4773–4776 (1996). [CrossRef] [PubMed]

25. L. Wu, S. He, and L. Chen, “On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials,” Optics Express **11**1283–1290 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-12831. [CrossRef] [PubMed]

6. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**3966–3969 (2000) [CrossRef] [PubMed]

20. G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B **67** 035109, (2003) [CrossRef]

24. V. A. Podolskiy, “Plasmon modes and negative refraction in metal nanowire composites,” Optics Express **11**735–745 (2003) [CrossRef] [PubMed]

## 2. Formulation

1. V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and µ,” Soviet Physics USPEKHI **10**509–514 (1968) [CrossRef]

*n*is the complex index of refraction, ε is the relative complex permittivity, µ is the relative complex permeability, and

*k*is the complex propagation constant. Breaking these into real and imaginary parts we can obtain

*ε*

_{2}, µ

_{2},

*n*

_{2},

*k*

_{1}and

*k*

_{2}are positive. Note that from Eq. (4), the sign of

*n*

_{1}depends on the sign of the quantity (ε

_{1}µ

_{2}+µ

_{1}ε

_{2}). The absorption per unit length in the material is

*L*

_{m}) where

## 3. Applications

9. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**77–79 (2001) [CrossRef] [PubMed]

*f*=10.5 GHz. We obtain Λ

_{m}=0.40 cm

^{-1}, λ

_{m}=0.78 cm,

*L*

_{m}=0.31. This indicates that about a third of the energy is lost per cm through the structure. This magnitude of loss would have to be considered in any application of left handed materials. In reference [9

9. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**77–79 (2001) [CrossRef] [PubMed]

*et al.*[22

22. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielan, “Experimental Verification and Simulation of Negative Index of Refraction Using Snell’s Law,” Phys. Rev. Lett. **90** 107401 (2003) [CrossRef] [PubMed]

*L*

_{a}and of

*L*

_{m}for their three structures shown in Table 1 below. Also shown is the resonance wavelength, λ′

_{o}; the wavelength, in air, λ

_{a}, at which

*L*

_{a}is a minimum; and the wavelength, in air, λ

_{am}, at which

*L*

_{m}is a minimum for each structure.

_{1}and

*n*

_{1}are negative for all three structures and µ

_{1}is negative for the first two.

## 4. Analysis

_{1}is negative only in the close vicinity of the resonance of the split-ring structures where µ

_{2}is large. To get some insight into this we will make some approximations. First, for all three structures |ε

_{1}|≫ε

_{2}. In this limit Eq. (7) becomes

*a*=

*ε*

_{1}

*µ*

_{1}/|

*ε*

_{1}

*µ*

_{1}|=±1, and

*x*=

*µ*

_{2}/|

*µ*

_{1}|. The loss per wavelength in the material becomes independent of the permittivity although, of course, the wavelength itself does not. In Eq. (9) the minimum of

*L*

_{m}occurs at the minimum of

*x*. From O’Brien and Pendry [16] the minimum of

*x*occurs at the frequency ω where

*L*

_{m}and

*λ*

_{am}to within about ±4%.

_{o}is the permeability of free space, and ω is the angular frequency of the radiation. The result is

*f*=

*πR*

^{2}/

*a*

^{2}, is the fill factor,

*R*is the radius, and

*d*is the width of the split ring. Equation (13) can be rewritten as

*δ*=(

*ρλ/πR*

_{o})

^{1/2}where λ is the free space wavelength of the radiation and

*R*

_{o}=377Ω is the impedance of free space. Equation (14) indicates that losses will become very large as

*d*and

*R*become comparable to the skin depth. Equation (14) can be rewritten as

*R*and

*d*are constrained by the equation π(

*R*+

*d*)

^{2}=

*b*

^{2}where

*b*is limited by the unit cell dimension, a, which, in turn, must be much less than the free-space wavelength. We can minimize Eq. (15) subject to this restriction. The result is

*R*=3

*d*=3

*b*/4√π. Inserting this into Eq. (15)

15. N. Garcia and N. Nieto-Vesperinas, “Left-Handed Materials Do Not Make a Perfect Lens,” Phys Rev. Lett. **88** 207403 (2002) [CrossRef] [PubMed]

_{o}′=6a and in our calculations. For these the minimum value of

*L*

_{m}occurs at λ≈0.97λ′

_{o}≈5.8

*a*. In reference [16] b=0.52a. For silver ρ=1.62×10

^{-8}Ωm and Eq. (16) becomes

_{o}′=6a. This scaling of λ

_{o}′ with lattice dimension can, in principle at least, be accomplished by adjusting the capacitance of the split ring. The results are shown in Fig. 1.

*L*

_{m}this well. It is surprising, however, that

*L*

_{a}also fits the approximation as well as it does. Notice that the values of

*L*

_{a}and

*L*

_{m}calculated for the optimized structures are considerably less than those calculated for the initial structures of O’Brien and Pendry, given in Table 1. In addition for all of the optimized structures, even for λ

_{o}′< 1µm, µ

_{1}and ε

_{1}are both negative as is

*n*

_{1}indicating that all the structures should result in left-handed propagation although the propagation is severely damped for the shorter wavelength structures. These results indicate that it might be possible to realize a useful LHM structure using this split-ring approach at wavelengths in the mid-infrared if one is willing to accept some modest losses. Much shorter wavelength structures appear likely to be too lossy to be useful.

## 5. Conclusions

## References and Links

1. | V. G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and µ,” Soviet Physics USPEKHI |

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

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEE Transactions on Microwave Theory and Techniques , |

4. | 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. |

5. | D. R. Smith and N. Kroll, “Negative Refractive Index in Left-Handed Materials,” Phys. Rev. Lett. |

6. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

7. | R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. |

8. | M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science |

9. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

10. | T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, and S. Schultz, “ |

11. | R. Marqués, F. Medina, and R. Rafi-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed materials,” Phys. Rev. B |

12. | S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys: Condens. Matter |

13. | 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 |

14. | P. M. Valanju, R. M. Walser, and A. P. Valanju. “Wave Refraction in Negative-Index Media: Always Positive and Very Inhomogeneous,” Phys. Rev. Lett. |

15. | N. Garcia and N. Nieto-Vesperinas, “Left-Handed Materials Do Not Make a Perfect Lens,” Phys Rev. Lett. |

16. | S. O’Brien and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic photonic crystals,” J. Phys: Condens. Matter |

17. | D. R. Smith, D. Schurig, and J. B. Pendry, “Negative refraction of modulated electromagnetic waves,” Appl. Phys. Lett. |

18. | L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, “Optomagnetic composite medium with conducting nanoelements,” Phys. Rev. B |

19. | J. Pacheco, T. M. Grzegorzyk, B.-I. Wu, Y. Zhang, and J. A. Kong, “Power Propagation in Homogeneous Isotropic Frequency-Dispersive Left-Handed Media,” Phys. Rev. Lett. |

20. | G. Shvets, “Photonic approach to making a material with a negative index of refraction,” Phys. Rev. B |

21. | J. B. Pendry and D. R. Smith, “Comment on “Wave Refraction in Negative-Index Media: Always Positive and Very Inhomogeneous,” Phys. Rev. Lett. |

22. | C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielan, “Experimental Verification and Simulation of Negative Index of Refraction Using Snell’s Law,” Phys. Rev. Lett. |

23. | S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in Media with a Negative Refractive Index,” Phys. Rev. Lett. |

24. | V. A. Podolskiy, “Plasmon modes and negative refraction in metal nanowire composites,” Optics Express |

25. | L. Wu, S. He, and L. Chen, “On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials,” Optics Express |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(160.4670) Materials : Optical materials

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 29, 2003

Revised Manuscript: September 5, 2003

Published: September 22, 2003

**Citation**

John Dimmock, "Losses in left-handed materials," Opt. Express **11**, 2397-2402 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-19-2397

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### References

- V. G. Veselago, �??The Electrodynamics of Substances with Simultaneously Negative Values of ε �?�and µ,�?? Soviet Physics USPEKHI 10 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, �??Extremely Low Frequency Plasmons in Metallic Mesostructures,�?? Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, �??Magnetism from Conductors and Enhanced Nonlinear Phenomena,�?? IEE Transactions on Microwave Theory and Techniques, 47 2075-2084 (1999). [CrossRef]
- 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 4184-4187 (2000). [CrossRef] [PubMed]
- D. R. Smith and N. Kroll, �??Negative Refractive Index in Left-Handed Materials,�?? Phys. Rev. Lett. 85 2933-2936 (2000). [CrossRef] [PubMed]
- J. B. Pendry, �??Negative Refraction Makes a Perfect Lens,�?? Phys. Rev. Lett. 85 3966-3969 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two dimensional, isotropic, left-handed metamaterial,�?? Appl. Phys. Lett. 78 489-491 (2001). [CrossRef]
- M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, �??Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,�?? Science 291, 849-851 (2001). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental Verification of a Negative Index of Refraction,�?? Science 292 77-79 (2001). [CrossRef] [PubMed]
- T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, and S. Schultz, �??Ab initio numerical simulation of left-handed metamaterials: Comparison of calculations and experiments,�?? J. Appl. Phys. 90 5419-5424 (2001). [CrossRef]
- R. Marqués, F. Medina, and R. Rafi-El-Idrissi, �??Role of bianisotropy in negative permeability and lefthanded materials,�?? Phys. Rev. B 65 144440 (2002). [CrossRef]
- S. O�??Brien and J. B. Pendry, �??Photonic band-gap effects and magnetic activity in dielectric composites,�?? J. Phys: Condens. Matter 14, 4035-4044 (2002). [CrossRef]
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, �??Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,�?? Phys. Rev. B 65 195104 (2002). [CrossRef]
- P. M. Valanju, R. M. Walser, and A. P. Valanju. �??Wave Refraction in Negative-Index Media: Always Positive and Very Inhomogeneous,�?? Phys. Rev. Lett. 88 187401 (2002). [CrossRef] [PubMed]
- N. Garcia and N. Nieto-Vesperinas, �??Left-Handed Materials Do Not Make a Perfect Lens,�?? Phys Rev. Lett. 88 207403 (2002). [CrossRef] [PubMed]
- S. O�??Brien and J. B. Pendry, �??Magnetic activity at infrared frequencies in structured metallic photonic crystals,�?? J. Phys: Condens. Matter 14 6393-6394 (2002).
- D. R. Smith, D. Schurig and J. B. Pendry, �??Negative refraction of modulated electromagnetic waves,�?? Appl. Phys. Lett. 81 2713-2715 (2002). [CrossRef]
- L. V. Panina, A. N. Grigorenko, and D. P. Makhnovskiy, �??Optomagnetic composite medium with conducting nanoelements,�?? Phys. Rev. B 66 155411 (2002). [CrossRef]
- J. Pacheco, T. M. Grzegorzyk, B.-I. Wu, Y. Zhang, and J. A. Kong, �??Power Propagation in Homogeneous Isotropic Frequency-Dispersive Left-Handed Media,�?? Phys. Rev. Lett. 89 257401 (2002). [CrossRef] [PubMed]
- G. Shvets, �??Photonic approach to making a material with a negative index of refraction,�?? Phys. Rev. B 67 035109, (2003). [CrossRef]
- J. B. Pendry and D. R. Smith, �??Comment on �??Wave Refraction in Negative-Index Media: Always Positive and Very Inhomogeneous,�?? Phys. Rev. Lett. 90 029703 (2003). [CrossRef] [PubMed]
- C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielan, �??Experimental Verification and Simulation of Negative Index of Refraction Using Snell�??s Law,�?? Phys. Rev. Lett. 90 107401 (2003). [CrossRef] [PubMed]
- S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, �??Refraction in Media with a Negative Refractive Index,�?? Phys. Rev. Lett. 90 107402 (2003). [CrossRef] [PubMed]
- V. A. Podolskiy, �??Plasmon modes and negative refraction in metal nanowire composites,�?? Optics Express 11, 735-745 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-735">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-735</a> [CrossRef] [PubMed]
- L. Wu, S. He, and L. Chen, �??On unusual narrow transmission bands for a multi-layered periodic structure containing left-handed materials,�?? Optics Express 11 1283-1290 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1283">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1283 </a> [CrossRef] [PubMed]

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