## Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial

Optics Express, Vol. 11, Issue 7, pp. 696-708 (2003)

http://dx.doi.org/10.1364/OE.11.000696

Acrobat PDF (1970 KB)

### Abstract

We have previously shown that a new class of Negative Refractive Index (NRI) metamaterials can be constructed by periodically loading a host transmission line medium with inductors and capacitors in a dual (high-pass) configuration. A small planar NRI lens interfaced with a Positive Refractive Index (PRI) parallel-plate waveguide recently succeeded in demonstrating focusing of cylindrical waves. In this paper, we present theoretical and experimental data describing the focusing and dispersion characteristics of a significantly improved device that exhibits minimal edge effects, a larger NRI region permitting precise extraction of dispersion data, and a PRI region consisting of a microstrip grid, over which the fields may be observed. The experimentally obtained dispersion data exhibits excellent agreement with the theory predicted by periodic analysis, and depicts an extremely broadband region from 960MHz to 2.5GHz over which the refractive index remains negative. At the frequency at which the theory predicts a relative refractive index of -1, the measured field distribution shows a focal spot with a maximum beam width under one-half of a guide wavelength. These results are compared with field distributions obtained through mathematical simulations based on the plane-wave expansion technique, and exhibit a qualitative correspondence. The success of this experiment attests to the repeatability of the original experiment and affirms the viability of the transmission line approach to the design of NRI metamaterials.

© 2003 Optical Society of America

## 1. Introduction

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

**E**,

**H**, and

**k**form a left-handed triplet. Veselago postulated several interesting phenomena associated with LHM, including the reversal of conventional refraction, Cherenkov radiation, the Doppler shift, and the focusing of cylindrical waves at a planar interface using flat lenses. Interest in LHM was revived in 1999 when an artificial dielectric was designed and verified in experiment to exhibit negative material parameters giving rise to a negative effective refractive index at microwave frequencies [2

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

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

6. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. on Microwave Theory and Tech. **50**, 2702–2712 (2002). [CrossRef]

8. A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. **92**, 5930–5935 (2002). [CrossRef]

## 2. Theory

8. A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. **92**, 5930–5935 (2002). [CrossRef]

*and ε*

_{P}*describe the intrinsic material parameters of the host transmission line medium, contributing positively to the equivalent parameters of the NRI medium, and the reactive inclusions C*

_{P}_{0}and L

_{0}provide the desired negative contribution that diminishes with frequency ω. Of course, such a mapping from circuit to field quantities also entails the specific geometry of the transmission line, the effect of which is lumped into the parameter

*g*of Eq. (1) through the ratio of the characteristic impedance of the transmission line to the intrinsic impedance of the bulk medium surrounding the network. Similar expressions were previously presented in [6

6. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. on Microwave Theory and Tech. **50**, 2702–2712 (2002). [CrossRef]

8. A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. **92**, 5930–5935 (2002). [CrossRef]

*g*parameter. The PRI parameters µ

*and µ*

_{P}*may therefore be obtained from the per-unit-length capacitance*

_{P}*C*and inductance

_{x}*L*of the transmission line segments comprising the host medium through the parameter

_{x}*g*, according to Eq. (2):

*C*is a correction factor in the extrapolation from one to two dimensions [6

_{x}6. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. on Microwave Theory and Tech. **50**, 2702–2712 (2002). [CrossRef]

**92**, 5930–5935 (2002). [CrossRef]

_{r}=2.94) substrate of height

*h*=1.524mm (60mil) and consists of a PRI region measuring 21×21 cells (105mm×105mm), and an adjacent NRI region measuring 21×40 cells (105mm×200mm). Both the PRI medium, and the NRI host medium, consist of a square grid of

*w*=400µm wide microstrip lines with

*d*=5mm period, for which the distributed parameters and characteristic impedance may be obtained using standard quasi-static formulas [9,10], given in Eqs. (3)–(5).

*, for which the intrinsic impedance may be represented as η*

_{eff}*, and the per-unit-length capacitance*

_{eff}*C*and inductance

_{x}*L*are obtained through the microstrip characteristic impedance

_{x}*Z*. Substituting the design values yields ε

_{0}*=2.11,*

_{eff}*g*=0.54419, ε

*=2ε*

_{P}*ε*

_{eff}*and µ*

_{0}*=µ*

_{P}_{0}, where we remind the reader that ε

*represents the effective permittivity surrounding a single microstrip line. In the NRI region, 5.6nH chip inductors are embedded (in shunt) into rectangular holes punched into the substrate at each cell site, and chip capacitors of 1pF are surface-mounted between gaps separating the cells. To maintain uniformity throughout, 2pF capacitors were placed at the array edges, followed by matching resistors.*

_{eff}*=2π×2.18GHz.*

_{0}*perfect*focusing is far more stringent, requiring that the PRI and NRI media be impedance matched at ω

*such that µ*

_{0}*(ε*

_{N}*)=-µ*

_{0}*and ε*

_{P}*(ω*

_{N}*)=-ε*

_{0}*. Many researchers have also noted the extreme sensitivity of this requirement, which may prove challenging for practical implementation (e.g., [12]). From the values chosen, it is apparent that we have relaxed this requirement so as to more easily investigate the salient features of focusing.*

_{P}### 2.1 Dispersion characteristics

*=2ε*

_{P}*ε*

_{eff}*and µ*

_{0}*=µ*

_{P}*, and*

_{0}*Z*is the characteristic impedance of the microstrip lines, as given by Eq. (4). The propagation constant is also approximated by the equivalent material parameters of Eq. (1) through the relation

_{0}14. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” IEEE Micro. and Opt. Tech. Lett. **31**, 129–133 (2001). [CrossRef]

*d*=π, described by the following quadratic equation:

*=2π×960MHz. Near the Bragg frequency, the phase delay through the transmission line component of the unit cell, θ, becomes small, and the solution is approximated by [6*

_{b}**50**, 2702–2712 (2002). [CrossRef]

*d*small), cannot predict the existence of this Bragg condition, and the associated dispersion curve diverges as ω→0. Consequently, the validity of Eq. (1) is limited to the interior LH region, before the stopband. The cutoff frequencies for the first stopband, however, are predicted both by Eq. (8) and Eq. (1), and are approximately given by [6

**50**, 2702–2712 (2002). [CrossRef]

*=2π×2.72GHz and ω*

_{C,1}*=2π×3.63GHz, respectively. Figure 2 also illustrates the dispersion of the PRI medium (dotted curve), which intersects the NRI dispersion curve near ω*

_{C,2}*=2π×2.18GHz, where it was predicted by Eqs. (6) and (7) that the condition*

_{0}*n*=-1 would be encountered.

_{REL}### 2.2 Plane-wave expansion analysis

*et al.*[15

15. J. A. Kong, B. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. **80**, 2084–2086 (2002). [CrossRef]

*d*is placed at

_{N}*x*=

*d*and excited by a vertical (

_{S}*y*-directed) line source located at

*x*=

*z*=0 inside a PRI medium. For

*n*=-1, the focal plane is located at

_{REL}*x*=2·

*d*. The resistive terminations

_{S}*Z*to the right of the NRI material are modeled by a third medium with positive parameters µ

_{term}*=µ*

_{T}*and ε*

_{0}*=ε*

_{T}*ε*

_{term}*, where ε*

_{0}*=(η*

_{term}*/*

_{0}*Z*)

_{term}^{2}, and all media are considered infinite in the transverse

*z*-direction. The plane-wave expansion method is then used to solve the problem of Fig. 3.

*i*={P,N,T} are represented as follows (note: a positive time-harmonic variation of the form exp(

*j*ω

*t*) is assumed):

*i*, and the positive root is chosen only for the NRI metamaterial. The resulting expression for the spectral components of the

*y*-directed electric field in the NRI metamaterial is

*d*=10 cells from the interface. Figure 4 shows the resulting vertical electric field magnitude and phase distributions at ω

_{S}*=2π×2.18GHz, the frequency at which Eqs. (6) and (7) predict*

_{0}*n*=-1.

_{REL}*x*=2·

*d*=cell 20), which have been indicated by a contour at the - 3dB level in the NRI medium, and the reversal of the concavity of the phase-fronts at the interface and at the focal plane. Although focusing is apparent, these results do not suggest

_{S}*perfect*focusing, since the PRI and NRI media are not impedance-matched at ω

*. That is, while µ*

_{0}*(ω*

_{N}*)ε*

_{0}*(ω*

_{N}*)=ε*

_{0}*µ*

_{P}*at 2.18GHz (the*

_{P}*n*=-1 condition), the material parameters have not been individually synthesized such that µ

_{REL}*(ω*

_{N}*)=-µ*

_{0}*and ε*

_{P}*(ω*

_{N}*)=-ε*

_{0}*, resulting in a beam characteristic similar to the results reported in [13]. We note that µ*

_{P}*(ε*

_{N}*) and ε*

_{0}*(ω*

_{N}*) can be individually synthesized to meet this constraint through the proper selection of the reactive inclusions*

_{0}*L*and

_{0}*C*in Eq. (1); however, the precision with which this is achieved is limited by device and process tolerances.

_{0}## 3. Experiment

*d*) along the central row (row 11) of the device, and is presented in Fig. 6 (only the LH passband is depicted; the experimental data within the stopband, and the subsequent higher-order band, is obscured due to strong mismatch losses at these frequencies). This dispersion relation reveals a well-defined region of backward-wave propagation extending from the Bragg frequency (960MHz) to approximately 2.5GHz that exhibits a distinct NRI characteristic [5,6

**50**, 2702–2712 (2002). [CrossRef]

*d*=5mm in the present case) often do not adequately sample the rapidly varying phase at these frequencies. The dispersion data of Fig. 6 indicates that this phenomenon becomes evident near 1.17GHz, where the guide wavelength is calculated to be 17.4mm, roughly equal to 3.5

*d*. Reference [16] has investigated the issue of granularity in the context of the transmission line method (TLM), and describes a similar cutoff of λ=4

*d*.

*n*=ϕ

*c*/ω

*d*, where

*c*is the speed of light in vacuum and ϕ is the measured average phase shift per unit cell β

*d*. In the most well defined region from 1.17GHz to 2.03GHz, the absolute refractive index ranges from approximately -14.76 to -2.63. Since the PRI region is expected to possess an absolute refractive index of √(2·ε

*)=+2.05, the corresponding relative refractive index*

_{eff}*n*varies from approximately -7.20 to -1.28. Using Eqs. (6) and (7), it was predicted that the device would encounter the condition

_{REL}*n*=-1 near ω

_{REL}*=2π×2.18GHz, which is indicated in Fig. 6 at the intersection between the theoretical NRI and PRI dispersion curves. However, this condition is achieved earlier, near ω′*

_{0}*=2π×2.09 GHz, approximately where the experimental NRI dispersion curve intersects with the theoretical PRI dispersion curve.*

_{0}*n*=-1. Giving further credibility to this observation is the fact that the absolute phase at the source plane (cell 0) is almost exactly restored at the image plane (cell 20) at this frequency. The magnitude and phase of the measured vertical electric fields over the structure at 2.09GHz are depicted in Fig. 8. The increased transmission at the focal plane, as well as the convergent progression of the wavefronts of Fig. 8 are indicative of focusing, and exhibit a qualitative correspondence with the field distribution at 2.18GHz predicted by the plane-wave expansion analysis (Fig. 4). The phase distributions between the two results, however, exhibit some discrepancy, which may be attributed to the many levels of simplification inherent in the plane-wave expansion theory used. Specifically, the theory employs homogeneous media in a purely two-dimensional geometry (i.e., infinite in the

_{REL}*y*- and

*z*-directions of Fig. 3), and therefore cannot account for the effect of periodicity, grid truncation, termination, and planar confinement evident in the experimental results. Moreover, the phase suffers from a greater sensitivity to these conditions than does the magnitude. However, at their respective frequencies, the phase distributions of the experiment and theory reveal guide wavelengths that are approximately equal (λ

_{NRI,experiment}=70mm and λ

_{NRI,theory}=67.6mm). Consequently, the beam widths of Fig. 4 and Fig. 8 at the -3dB level may be compared; the maximum beam widths in millimeters are found to be 32mm in experiment and 23mm in the theory, corresponding to 0.46λ

_{NRI,experiment}and 0.34λ

_{NRI, theory}, respectively.

*. These phenomena are reminiscent of the surface modes described by Ruppin [17*

_{0}17. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter **13**, 1811–1819 (2001). [CrossRef]

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

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

*absolute*refractive index at 2.4GHz is almost exactly -1, suggesting the strong possibility that in this frequency region, extending into the impending stopband, fast waves are excited at the NRI-air surface boundary, and the structure radiates. The entire region over which the refractive index remains negative, from approximately 960MHz to 2.5GHz, corresponds to a bandwidth that may be conservatively placed at 85%.

*n*=-1, the focus appears. The focal spot recedes slowly to the interface as the frequency is increased, a phenomenon predicted by geometrical optics for NRI media [6

_{REL}**50**, 2702–2712 (2002). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. |

2. | J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. on Microwave Theory and Tech. |

3. | 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. |

4. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

5. | A. K. Iyer and G. V. Eleftheriades, “Negative refractive index metamaterials supporting 2-D waves,” in MTT-S International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, Seattle, 2002) pp. 1067–1070. |

6. | G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. on Microwave Theory and Tech. |

7. | A. Grbic and G. V. Eleftheriades, “A backward-wave antenna based on negative refractive index L-C networks,” in IEEE Int. Symp. Ant. and Propag. (Institute of Electrical and Electronics Engineers, San Antonio, 2002) pp. 340–343. |

8. | A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. |

9. | R. E. Collin, |

10. | D. M. Pozar, |

11. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

12. | N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” in Proc. 2 |

13. | R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E64, 056625: 1–15. |

14. | I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” IEEE Micro. and Opt. Tech. Lett. |

15. | J. A. Kong, B. Wu, and Y. Zhang, “Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,” Appl. Phys. Lett. |

16. | W. J. R. Hoefer, “The Transmission Line Matrix (TLM) Method,” in |

17. | R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(100.3020) Image processing : Image reconstruction-restoration

(110.2990) Imaging systems : Image formation theory

(220.3630) Optical design and fabrication : Lenses

(350.4010) Other areas of optics : Microwaves

**ToC Category:**

Focus Issue: Negative refraction and metamaterials

**History**

Original Manuscript: January 28, 2003

Revised Manuscript: March 24, 2003

Published: April 7, 2003

**Citation**

Ashwin Iyer, Peter Kremer, and George Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express **11**, 696-708 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-696

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### References

- V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and μ,�?? Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robins, W. J. Stewart, �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. on Microwave Theory and Tech. 47, 2075-2084 (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, �??Composite medium with simultaneously negative permeability and permittivity,�?? Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, S. Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77-79 (2001). [CrossRef] [PubMed]
- A. K. Iyer, G. V. Eleftheriades, �??Negative refractive index metamaterials supporting 2-D waves,�?? in MTT-S International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, Seattle, 2002) pp. 1067-1070.
- G. V. Eleftheriades, A. K. Iyer, P. C. Kremer, �??Planar negative refractive index media using periodically L-C loaded transmission lines,�?? IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002). [CrossRef]
- A. Grbic, G. V. Eleftheriades, �??A backward-wave antenna based on negative refractive index L-C networks,�?? in IEEE Int. Symp. Ant. and Propag. (Institute of Electrical and Electronics Engineers, San Antonio, 2002) pp. 340-343.
- A. Grbic, G. V. Eleftheriades, �??Experimental verification of backward-wave radiation from a negative refractive index metamaterial,�?? J. Appl. Phys. 92, 5930-5935 (2002). [CrossRef]
- R. E. Collin, Foundations for Microwave Engineering, 2nd Ed. (McGraw-Hill, Inc., Toronto, 1992).
- D. M. Pozar, Microwave Engineering, 2nd Ed. (John Wiley & Sons, Toronto, 1998).
- J. B. Pendry, �??Negative refraction makes a perfect lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- N. Fang, X. Zhang, �??Imaging properties of a metamaterial superlens,�?? in Proc. 2nd IEEE Conference on Nanotechnology (Institute of Electrical and Electronics Engineers, Washington D.C., 2002) pp. 225-228.
- R. W. Ziolkowski, E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625: 1-15.
- I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, S. Ilvonen, �??BW media �?? media with negative parameters, capable of supporting backward waves,�?? IEEE Micro. Opt. Tech. Lett. 31, 129-133 (2001). [CrossRef]
- J. A. Kong, B. Wu, Y. Zhang, �??Lateral displacement of a Gaussian beam reflected from a grounded slab with negative permittivity and permeability,�?? Appl. Phys. Lett. 80, 2084-2086 (2002). [CrossRef]
- W. J. R. Hoefer, �??The Transmission Line Matrix (TLM) Method,�?? in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, ed. (John Wiley & Sons, Toronto, 1989).
- R. Ruppin, �??Surface polaritons of a left-handed material slab,�?? J. Phys.: Condens. Matter 13, 1811-1819 (2001). [CrossRef]

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