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

Ashwin K. Iyer, Peter C. Kremer, and George V. Eleftheriades

Open Access Open Access

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

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Supplementary Material (2)

Media 1: AVI (1700 KB)     
Media 2: AVI (1657 KB)     

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Figures (8)
Fig. 1.
Fig. 1. Unit cell for the 2-D transmission line NRI metamaterial.

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Fig. 2.
Fig. 2. Dispersion relation for designed NRI medium as obtained through periodic analysis (dashed curve) and using the equivalent material parameters (solid curve), and dispersion relation for PRI medium (dotted curve). The NRI dispersion curves indicate a Left-Handed (LH) passband enclosed by the Bragg frequency ω b and stopband cutoffs ω C,1 and ω C,2 , and the intersection of the NRI and PRI dispersion curves indicate the frequency ω 0 at which nREL =-1.

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Fig. 3.
Fig. 3. PRI/NRI metamaterial interface examined by the plane-wave expansion theory. The arrows depict the wavevectors in each medium.

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Fig. 4.
Fig. 4. Vertical electric field distribution over a PRI/NRI interface (PRI: cells -10 to 10, NRI: cells 11 to 50) predicted by plane-wave expansion analysis at 2.18GHz (normalized individually in the PRI and NRI regions to their maximum respective amplitudes) illustrating focusing

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Fig. 5.
Fig. 5. Experimental prototype. The PRI region measures 21×21 cells (105mm×105mm), and the adjacent NRI region measures 21×40 cells (105mm×200mm). The inset magnifies a single NRI unit cell, consisting of a microstrip grid loaded with surface-mounted capacitors and an inductor embedded into the substrate at the central node. The near-field detecting probe is also depicted, and the arrow indicates the location of the vertical excitation probe.

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Fig. 6.
Fig. 6. Experimentally obtained NRI dispersion relation (solid curve) indicating the Bragg frequency at 960MHz and a well-defined NRI region extending to approximately 2.5GHz (data obtained for region βd=-π to βd=0 and reflected in the βd=0 axis for the space-reversed solution). Also depicted are the theoretical NRI (dashed curve) and PRI (dotted curve) dispersion relations. The intersection of the PRI dispersion with the experimental NRI dispersion at ω′ 0 differs slightly from the predicted intersection at ω 0 .

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Fig. 7.
Fig. 7. Experimental data showing phase profile in the PRI and NRI regions (PRI: cells -10 to 10, NRI: cells 11 to 50) along the central row of the experimental device at 2.09GHz

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Fig. 8.
Fig. 8. Experimental data showing field magnitude and phase distributions in PRI and NRI regions at 2.09GHz (PRI: cells -10 to 10, NRI: cells 11 to 50). (1.68MB movie showing experimental field magnitude distributions from 1.0GHz to 2.5GHz; 1.62MB movie showing experimental field phase distributions from 1.0GHz to 2.5GHz).

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Equations (17)

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μ N ( ω ) = μ p − 1 g ω 2 C 0 d , ε N ( ω ) = ε p − g ω 2 L 0 d .
ε P = 2 C x · g , μ P = L x g ,
ε eff = ε r + 1 2 + ε r − 1 2 1 1 + 12 h w , η eff = η 0 ε eff , η 0 = 377 Ω
Z 0 = η eff g , g = 1 2 π ln ( 8 h w + w 4 h )
C x = 2 π ε 0 ε eff ln ( 8 h w + w 4 h ) = ε 0 ε eff g , L x = Z 0 2 · C x = μ 0 g
μ N ( ω 0 ) ε N ( ω 0 ) = μ P ε P .
ω 0 = ( L 0 · 2 ε eff ε 0 d g + C 0 · g μ 0 d ) − 1 2 ,
cos ⁡ βd = cos ⁡ θ − 1 2 ω 2 L 0 C 0 cos ⁡ 2 θ 2 + 1 2 ω ( 1 C 0 Z 0 + 1 L 0 Y 0 ) sin ⁡ θ
β = − ω μ N ( ω 0 ) ε N ( ω 0 ) .
A ω b 4 + B ω b 2 + 1 = 0 ,
A = L 0 C 0 μ P ε P d 2 , B = − ( L 0 ε P d g + C 0 g μ P d + 4 L 0 C 0 ) ,
ω b = 1 2 L 0 C 0 .
ω c , 1 = 1 C 0 g μ P d , ω c , 2 = 1 L 0 ε P d g ,
E y , i ( x , z ) = 1 2 π ∫ − ∞ ∞ E ˜ y , i ( x , k x , i ) e − jk z , i z dk z , i ,
k x , i = ± j k z , i 2 − k i 2 , k i 2 = ω 2 ε i μ i .
E ˜ y , N = 2 E ˜ y , P ∣ x = 0 e − j k x , P d S { ( τ B + 1 ) e − j k x , N ( x − ( d S + d N ) ) + ( τ B − 1 ) e + j k x , N ( x − ( d S + d N ) ) ( τ A + 1 ) ( τ B + 1 ) e + j k x , N d N − ( τ A − 1 ) ( τ B − 1 ) e − j k x , N d N }
τ A = k x , N μ P k x , P μ N , τ B = k x , N μ T k x , T μ N , E ˜ y , P ∣ x = 0 = − ωμ P 2 k x , P .
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