Does a Negative Refractive Index always result in Negative Refraction? – Effect of Loss

Vasundara V. Varadan, and Liming Ji

Microwave and Optics Laboratory for Imaging & Characterization Department of Electrical Engineering, University of Arkansas, Fayetteville

Abstract – The phenomenon of negative refraction has been

demonstrated experimentally and by numerical simulation as-suming Drude and Lorentz models for the permittivity and per-meability. It has been assumed that a negative refractive index results in negative refraction and hence will lead to a variety of exciting applications for metamaterials. Loss cannot be avoided in real metamaterials and in this paper, we analyze the effect of loss on negative refraction using experimentally extracted data for the permittivity, permeability and refractive index of a com-bined wire-Split Ring Resonator sample and for a sample with alternately oriented split ring resonators. Both samples are very lossy in the plasmonic resonance region and both display a nega-tive refractive index. We show that obliquely incident waves on such samples may not always lead to negative refraction due to the effect of losses in the material.

Index terms – negative refractive index, loss, negative refraction, metamaterial

I. INTRODUCTION

Metamaterials consisting of arrays of split ring resonators (SRRs) and metal wires have been used to demonstrate the phenomenon of negative refraction as envisaged by Veselago [1]. Following this pioneer work, the science and engineering research community has been inspired and motivated to pur-sue different aspects of research pertaining to negative index and other interesting phenomena relating to metamaterials [2-4]. Negative index metamaterials reported in [1, 5] has the periodicity of one SRR and one metal wire and the cut-off frequency of the metal wire employed is significantly higher than the resonance frequency of the SRR. As the effective properties of arrays of SRRs or metal wires depend on geo-metrical configurations of metal structures as well as the pe-riodicity of the arrays [5,6,7], a negative permeability or per-mittivity medium can be achieved by tailoring those parame-ters. In [8] it has been shown that both electric and magnetic resonances can be realized by appropriately orienting the SRRs. It follows that negative index metamaterials may be realized by a proper design of those parameters. Most papers dealing with theory or numerical simulations assume that only the permittivity or the permeability displays dispersive behav-ior. Thus a Lorentz model is assumed for the permeability in order to realize an artificial magnetic medium, but the dielec-tric permittivity is assumed to be constant [7]. This is not pos-sible in practical samples for which both the permittivity and permeability are dispersive simultaneously. Losses are ignored

in many theoretical and numerical treatments. This is also not true in practically realizable samples.

Using this paradigm, we conducted a series of experiments on metamaterial samples combining (i) SRRs and wire strips in various arrangements and periodicities and (ii) alternate orthogonally oriented SRRs in a periodic array. In each case, we used the fully calibrated free space measurement system to accurately extract the complex permittivity and permeability and hence the refractive index of the samples for normally incident waves. Both samples are lossy in the frequency band where the refractive index is negative. We next analysed obliquely incident waves on such samples that have a negative refractive index but are lossy and conclude that negative re-fraction is a sensitive function of the angle of incidence and loss in the material and that negative refraction may or may not be possible even though the refractive index is negative.

II. SAMPLES AND MEASUREMENTS

SRR shapes employed in this paper were printed on a di-electric substrate by conventional wet lithography using the geometrical parameters provided in [8]. Each square cell has a lattice constant of 5mm and the thickness of metallization is 17 microns. The substrate is then cut into strips, which then can be supported by the sample holder. Each strip is fitted into a groove in a picture frame holder [8] so that the pattern printed on the dielectric substrate FR4 is not fully visible and we only see the edge of the FR4 strips, Fig. 1a. Each strip con-sists of more than 20 SRRs and 45 strips were put into sample holder with spacing between each strip 2.5 mm. The lateral period is 10 mm and period in the direction of wave propaga-tion is 5 mm. The wire strips are 1.2 mm wide and the SRR are vertically spaced 5 mm apart and have their gaps perpen-dicular to the incident electric field and parallel to the metal strips. The magnetic field is perpendicular to the plane of the SRR and the wave vector is parallel to the plane of the SRR (Fig. 1a). The metal strip has the dimension 0.96×5×115 mm3. The samples were then placed on a sample holder for S-parameters measurement using a free-space measurement sys-tem with horn lens antennas that produce Gaussian beam illu-mination. There are approximately 450 SRRs within the illu-minating beam. First TRL calibration is implemented, the amplitude and phase of the complex S-parameters S11 and S21 are measured and the complex properties are extracted as described in [8] using

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(d) Figure 1: Wire trip – SRR Sample (a) Geometry and arrangement; (b) Measured S-parameters; (c) Power Absorption; and (d) Extracted complex wavenumber

(c)

(b)

(a)

Fresnel formulae for the assumed effectively homogeneous medium. Our experimental studies indicate that the periodicity of metal strips has significant effects on the frequency re-sponse of metamaterials and the periodicity of metal strip- SRR sample was adjusted so that the plasma frequency of the wire strip only sample is higher than the plasmon resonance frequency of the SRR sample. In Fig. 1b we plot the magnitude of the S-parameter, the measured phase of S11 and S21 is not shown but is used for the extraction of material properties. We note that the |S21|min occurs at a lower fre-quency than the |S11|min and this is in contrast to the pure SRR sample [8]. The phase of both S11 and S21 was observed to be dispersive, in [7] for the SRR only samples, only the phase of S21 is dispersive.

In Fig. 1c we have plotted the power absorption in the sam-ple computed as 1-|S11|

2-|S21|2. We observe the strong peak at

10.2 GHz. As described in [8], the S-parameters were used to extract the complex wavenumber and complex impedance of the effectively homogeneous medium. The complex wavenumber is plotted as a function of frequency in Fig, 1d. We note that the real part of the wavenumber is negative in the 10.0-10.4 GHz frequency range. The imaginary part of the wavenumber is very high (large attenuation) below 10.0 GHz and then rapidly stabilizes to a small low value above the re-gion of negative refractive index. This is the first complete measurement of all the properties of the combined wire-SRR medium displaying negative refractive index.

In Fig. 2a, we present the geometry of the sample for a fully alternate arrangement of orthogonally oriented SRRs, where adjacent SRRs have their gaps parallel and perpendicular to the incident E field. The amplitude of the S-parameters is shown in Fig. 2b and the power absorption in Fig. 2c. We no-tice the very high absorption at the resonance frequency and this is because the S11 and S21 minima occur at the same fre-quency. In Fig. 2d, the extracted complex wavenumber is plot-ted and it shows a very strong negative behavior at 10.2 GHz.

We conclude this section by observing that in practical metamaterials, negative refractive index is accompanied by very high power absorption. Further we have shown that nega-tive refractive index can be realized with appropriately ori-ented SRRs as hinted in [7] using the metasolenoid. To our knowledge, this is the first experimental extraction of negative refractive index at microwave frequencies.

Using full wave simulation, we can compare the measured S-parameters. There is excellent agreement of both the ampli-tude and phase of the measured S-parameters. Due to the high loss at resonance, it is not possible to measure the S-parameters of multiple layers of this material to prove that homogenization is possible, i.e. different thickness samples will lead to the same complex wavenumber. But the phe-nomenon of negative properties is essentially due to plasmonic resonance and occurs only in a small frequency band. We may question the applicability of effective medium theory in the resonance region. Outside the resonance region, the medium behaves just like an ordinary, non-dispersive, low loss dielec-tric. Concepts of negative properties applied in the resonance

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region are simply a tool to understand and explain the meas-ured S-parameters and the dispersion of the phase of the S-parameters. It is in this spirit, that we present the complex wavenumber for these samples. This will be more fully elabo-rated in a future publication.

III. OBLIQUE INCIDENCE OF TM WAVES ON A LOSSY NEGA-

TIVE INDEX MATERIAL

The interface of two infinite materials at z=0 is considered. Plane EM waves are incident at an angle θ0 to the unit normal to the interface with TM polarization (H field perpendicular to plane of incidence, the x-z plane). The material for z>0 is con-sidered to be a lossy negative index material (NIM) with n=n’+jn”. The boundary value problem of reflection and transmission of waves at such an interface is studied. We refer to the excellent treatment given in [9]. Phase matching of the fields along the boundary or Snell’s law requires k0 sinθ0 = kx = k sinθt where k = k '− jk" is the wavenumber

for z>0 and θt is transmitted wave phase angle. It is conven-ient to write the phase angle as

sinθt =k0

k "+ jk 'sinθ0; 

cosθt = 1+ (k0 sinθ0 )2

(k "+ jk ')2 = A + jB

(1)

We note that this is the phase angle and not the angle of re-fraction or the angle that the transmitted field makes with the unit normal. The transmitted magnetic field can be written as

H = yHt e− j(k '− jk ")(x sinθt+z cosθt ) =  yHt e

−α zze− j(kxx+kzz) (2) In Eq. (2), kx = k0 sinθ0; kz = k ' A + k " B; α z = k " A − k ' B . The refraction angle of the transmitted field, ψ can be written as

cosψ =kz

k= k ' A + k "B

(k0 sinθ0 )2 + (k ' A + k "B)2

(3)

For negative refraction, we require that ψ>90° whereas for a negative refractive index, we require that kz<0. This is a very important difference that has been overlooked.

Further it can be shown by analysing the Poynting vector, that for the case of TM polarization for the material to be an NIM, the real part of the permeability must be less than zero, Re(µ)<0. For the TE case, the real part of the permittivity should be negative in order to have the power flowing in the direction of wave propagation. It is not necessary for both permittivity and permeability to be negative to realize a NIM. This has been observed by others also.

For the samples measured in the previous section, the loss and hence B is quite high and this plays a very important role in determining whether ψ > 90° for a given θ0 even though kz<0. We will present the refraction angle for various angles of incidence for the two samples we have characterized and discuss the conditions for negative refraction. This is not shown due to lack of space.

(d)

(c)

(b)

(a)

Figure 2: Alternate orthogonal SRR lattice – (a) Geometry and ar-rangement; (b) Measured S-parameters; (c) Power Absorption; and (d) Extracted complex wavenumber

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IV. CONCLUSIONS

We have shown the first detailed measurements of the com-plex properties of combined SRR-wire media and also or-thogonal arrangements of SRRs and show that both have nega-tive refractive index in a limited frequency range. We have then discussed oblique angles of incidence in a lossy medium and discussed the difference between the phase angle and the angle that the refracted field makes with the wave normal. We conclude that loss in the NIM can significantly affect negative refraction and inhibit it for certain angles of incidence. This is the first detailed experimental and analytical study of the dif-ference between the phase angle and the refraction angle in a lossy metamaterial.

REFERENCES

[1] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184, 2000.

[2] G.W. Milton and N.A. Nicorovici, “On the cloaking effects associated with anomalous localized resonances”, Proc.Roy.Soc.A 462, pp. 3027-3059, 2006.

[3] J. B. Pendry, D. Schurig & D. R. Smith,” Controlling electro-magnetic fields”, Science, 312, pp. 1780-1782, 2006.

[4] D. Schurig, J. J. Mock,1 B. J. Justice, S. A. Cummer, J. B. Pen-dry, A. F. Starr, D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies”, Science, 314, pp. 977-980, 2006.

[5] K. Aydin and E. Ozbay, “Negative refraction through an imped-ance-matched left-handed metamaterial slab”, J. Opt. Soc. Am. B 23, 415, 2006.

[6] B. Sauviac, C. R. Simovski, and S. A. Tretyakov, “Double Split-Ring Resonators: Analytical Modeling and Numerical Simula-tions”, Electromagnetics 24, 317-338, 2004.

[7] S. Maslovski, P. Ikonen, I. Kolmakov, and S. Tretyakov, “Ar-tificial magnetic materials Based on the new magnetic particle: Metasolenoid”, Progress In Electromagnetics Research, PIER 54, 61–81, 2005.

[8] V.V. Varadan and A.R. Tellakula, "Measurement of the Com-plex Permittivity, Permeability and Refractive Index in Metama-terials Composed of Discrete Split Ring Resonators in the 8-26 GHz range”, Journal of Applied Physics, Vol. 100, 034910, 2006.

[9] C. T. A. Johnk, “Electromagnetic Fields and Waves”, John Wiley(1988), pp. 595-598.

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