Demonstration of negative refraction of microwaves

Demonstration of negative refraction of microwaves Maria C. Velazquez-Ahumada, Manuel J. Freire,a兲Jose M. Algarin, and Ricardo Marques Department of Electronics and Electromagnetism, University of Seville, 41012 Seville, Spain 共Received 30 June 2010; accepted 7 December 2010兲 An experimental setup to demonstrate negative refraction is described. A simple method for designing and fabricating a metamaterial with negative refractive index at microwave frequencies is discussed. The metamaterial is made of a multilayer planar arrangement of flat unit cells. A prism was fabricated and used to demonstrate negative refraction at the prism-air interface. The prism is designed for demonstrations and works at the frequency of commercial microwave transmitters and receivers. © 2011 American Association of Physics Teachers. 关DOI: 10.1119/1.3537122兴 I. INTRODUCTION In the well-known law of refraction in optics, the angles of incidence and refraction are measured on different sides from the normal to the boundary between two ordinary isotropic media 关see Fig. 1共a兲兴. In recent years, a new class of materials, called metamaterials,1have appeared which do not obey the law of refraction. Instead, at the boundary between an ordinary medium and a metamaterial, the angles of incidence and refraction can be on the same side from the normal 关see Fig. 1共b兲兴. This phenomenon is known as negative refraction and is a property of materials for which both the relative permittivity ␧rand the relative permeability ␮ rare both negative.2Metamaterials are composite materials with effective values of ␧rand ␮ rdetermined by their structure rather than by the intrinsic properties of the material components 共which are conventional conductors and dielectrics兲. In this paper, an experimental setup for demonstrating negative refraction is discussed. The demonstration uses electromagnetic waves with a frequency in the microwave band. A prism of metamaterial is designed and fabricated to operate in combination with commercial microwave transmitters and receivers commonly used in physics laboratories. II. THEORY Metamaterials with ␧r⬍0 and ␮ r⬍0 are also known as left-handed materials or negative refractive index materials, in contrast to ordinary materials which are right handed. The reason for labeling ␧r⬍0 and ␮ r⬍0 media as left-handed media can be briefly explained as follows.2Consider the Maxwell equations, ⵜ⫻E=−i ␻ ␮ H,共1兲 ⵜ⫻H=i ␻ ␧E,共2兲 where ␧=␧r␧0, ␮ = ␮ r ␮ 0, and ␧0and ␮ 0are the dielectric permittivity and magnetic permeability in vacuum. For the plane waves E=E0exp共−ik·r+i ␻ t兲and H=H0exp共−ik·r +i ␻ t兲, Eqs. 共1兲and 共2兲reduce to k⫻E= ␻ ␮ H,共3兲 k⫻H=− ␻ ␧E.共4兲 We see that for positive ␧and ␮ ,E,H, and kform a righthanded orthogonal system of vectors 关see Fig. 2共a兲兴. However, if ␧r⬍0 and ␮ r⬍0, then Eqs. 共3兲and 共4兲can be rewritten as k⫻E=− ␻ 兩 ␮ 兩H,共5兲 k⫻H= ␻ 兩␧兩E,共6兲 showing that E,H, and kform a left-handed triplet, as illustrated in Fig. 2共b兲. The main implication of this analysis is backward-wave propagation. The direction of the time-averaged flux of energy is determined by the real part of the Poynting vector, S=1 2E⫻Hⴱ,共7兲 which is unaffected by a simultaneous change of sign of ␧r and ␮ r. Thus, E,H, and Sstill form a right-handed triplet in a left-handed medium. In such media, the energy and wave fronts travel in opposite directions. Now consider refraction at the interface between an ordinary and a left-handed medium. The boundary conditions require the continuity of the tangential components of the wave vector along the interface. From the backward propagation in the left-handed region it follows that, unlike in ordinary refraction, the angles of incidence and refraction must have opposite signs. This effect is illustrated in Fig. 3. From the continuity of the tangential components of the wave vectors of the incident and refracted rays, it follows that 共see Fig. 3兲 sin ␪ 1 sin ␪ 2 =−兩k2兩 兩k1兩⬅n2 n1 ⬍0, 共8兲 which is the well-known law of refraction. In Eq. 共8兲n1and n2are the refractive indices of the ordinary and the lefthanded medium, respectively. If n1⬎0, it follows that n2 ⬍0 from Eq. 共8兲. That is, the sign of the square root in the refractive index definition must be negative,2 n⬅−冑␧r ␮ r⬍0. 共9兲 For this reason, left-handed media are also referred to as “negative refractive index” or “negative refractive” media.1 III. METHODOLOGY AND RESULTS A common way to demonstrate refraction is to form a prism, shine a beam through it, and observe the deflection of the beam on the other side. Waves enter the prism through one of the interfaces at normal incidence and strike the opposite interface at an oblique angle. For a prism made of ordinary or right-handed material this angle is positive 关see Fig. 4共a兲兴and is negative for a prism of left-handed material 关see Fig. 4共b兲兴. Negative refraction of waves in the micro349 349Am. J. Phys. 79 共4兲, April 2011 http://aapt.org/ajp © 2011 American Association of Physics Teachers wave range is demonstrated in an experiment where a microwave beam is refracted through a left-handed prism of a fabricated metamaterial. Negative refraction of microwaves was first shown experimentally by Shelby et al.3for a prism consisting of a threedimensional array of metallic wires and metallic split-ring resonators. Split-ring resonators provide a strong magnetic response and have been well studied 共see, for example, Ref. 1and references therein兲. Arrays of wires have also been analyzed for designing metamaterials with specific electric behavior.1In the metamaterial prism considered in Ref. 3the array of wires had ␧r⬍0, in contrast to the magnetic response of the split-ring resonators which was ␮ r⬍0. We will fabricate a metamaterial with a simpler structure. Because the prism is illuminated by linearly polarized plane waves coming from a microwave source, it is not necessary to have an isotropic material with scalar ␧r⬍0 and ␮ r⬍0. It is sufficient to fabricate an anisotropic material with ␧r⬍0in the direction of the electric field Eand ␮ r⬍0 in the direction of the magnetic field H. A simple way to obtain ␧r⬍0 in the direction of Eis to pile metallic plates which are parallel to the direction of E, so that each pair of closed plates constitutes a parallel plate waveguide.4The propagation constant ␤ of the fundamental transverse electric 共TE1兲or the magnetic 共TM1兲mode in a parallel plate waveguide is given by4 ␤ =冑k0 2− 冉 ␲ d 冊 2 ,共10兲 where k0is the free-space wave number, dis the distance between the plates, and ␲ /dis the cutoff wave number. The effective relative permittivity of this waveguide is4 ␧r= 冉 ␤ k0 冊 2 =1− 冉 ␲ dk0 冊 2 =1− 冉 c 2df 冊 2 ,共11兲 where fis the operating frequency, cis the speed of light in vacuum, and c/2drepresents the cutoff frequency. By choosing fto be less than the cutoff frequency, it is possible to obtain ␧r⬍0. To obtain ␮ r⬍0 in the direction of the Hfield, a two-dimensional planar array of split-ring resonators perpendicular to the Hfield is introduced between the plates. The fabricated prism consists of a multilayered planar arrangement of flat unit cells as shown in Fig. 5共a兲. Inside each cell ␧r⬍0 in the direction of Eand ␮ r⬍0 in the direction of H. Figure 5共b兲shows a sketch of one flat unit cell consisting of an array of split-ring resonators placed between two foam layers 共Rohacell 51 HF兲, 6 mm of thickness, and permittivity close to unity. The two foam layers are shielded by thin metallic plates at the top and the bottom 共the metal plates were made using the copper metallization of thin FR4 circuit boards兲. The thickness of each flat unit cell is 12 mm. The operating frequency of 10.5 GHz is fixed by the microwave transmitter and receiver used in the demonstration. For this frequency, Eq. 共11兲gives ␧r=−0.4 for d=12 mm. The geometry and dimensions of the array of split-ring resonators are shown in Fig. 5共c兲. We have developed models for making uniform metamaterials.5With the help of this model, which takes into account the resonance frequency of the split-ring resonators, dimensions, mutual couplings, and periodicity of the array, a desired magnetic response can be obtained. As a simple rule of thumb, to obtain ␮ r⬍0ata desired frequency, the split-ring resonator has to be designed to resonate at a frequency slightly below the operating frequency. In the device used in this paper, the resonance frequency of the fabricated split-ring resonators has an average value 共due to tolerances in the photoetching process兲of 10.35 GHz. Taking into account this resonance frequency Fig. 1. Refraction at the boundary between 共a兲two ordinary media or righthanded media and between 共b兲an ordinary medium and a left-handed medium. E H k S E H k S R ight-handed triplet Left-handed triple t ( a )( b ) Fig. 2. Illustration of the system of vectors E,H,k, and Sfor a plane wave in 共a兲an ordinary medium and 共b兲a left-handed medium. S 1 k 2 q 2 q 1 k 1 S 2 Fig. 3. Demonstration of the negative refraction between ordinary 共1—white兲and left-handed 共2—gray兲media. The Poynting and wave vectors at each media are labeled as S1,S2,k1, and k2, respectively. Negative refraction arises from the continuity of the components of the wave vectors, k1and k2, parallel to the interface and from the fact that rays propagate along the direction of energy flow. That is, rays must be parallel to the Poynting vectors S1and S2. Right-handed prism qiqt qiqt Left-handed prism qiqt (a) (b) Fig. 4. Refraction through a prism. The angle of refraction is positive for a prism made of 共a兲an ordinary or right-handed material and 共b兲is negative for a prism made of left-handed material or metamaterial. 350 350Am. J. Phys., Vol. 79, No. 4, April 2011 Velazquez-Ahumada et al. and the dimensions indicated in Fig. 5共c兲, our model gives ␮ r=−0.75 at the working frequency of 10.5 GHz.5 The values of ␧r=−0.4 and ␮ r=−0.75 give a negative index of refraction equal to ⫺0.55. The dimensions of the split-ring resonators were chosen with the help of freely available software.6The geometry shown in Fig. 5共c兲was chosen for its symmetry,7which ensures a purely magnetic response, so that the effective ␧rprovided by the metal plates is not affected by the presence of the split-ring resonators. The split-ring resonators were photoetched in an ARLON dielectric substrate LX02003311 with dielectric constant of 2.33 and dielectric thickness of 0.508 mm. A metamaterial consisting of a multilayered structure of three unit cells was fabricated, and a 20° prism was cut from it. The prism was placed in a slot made in a polyvinyl chloride 共PVC兲mounting base, and the base was placed between a microwave transmitter and a receiver. Figure 6shows a photograph of the setup. In the setup the transmitter is fixed and is normal to the output interface of the prism; the receiver can be rotated around the prism by means of a goniometer. The microwave transmitter and receiver were 10.5 GHz microwave horns 共PASCO兲. The metamaterial was designed to work at this frequency, but it can be easily designed to work at other frequencies by changing the resonance frequency of the split-ring resonators and the distance between the metal plates. The receiving horn was placed on the goniometer arm at a distance from the mounting base larger than ten wavelengths 共⬇30 cm兲to ensure that the wave at the receiving horn was approximately planar. The PVC mounting base was covered with adhesive copper foil, and pyramidal foam absorber was placed at both sides. This foam absorber prevents the formation of standing wave patterns between the horns and the mounting base. If standing wave patterns are present, the signal in the receiver would oscillate with the position of the receiver along the goniometer arm. A second 20° PVC prism was also fabricated for comparison purposes. Figure 7shows the measured radiation pattern for both prisms. The measurements were obtained by rotating the receiver between 50° and ⫺50° in steps of 5° and recording the voltage that was measured by a voltmeter connected to this receiver8共see the photograph in Fig. 6兲. The voltage is Metal plates Foam slabs Printed-circuit board LH Prism 2d=12 mm (d=6 mm ) r =1.63 mm ext g=0.2 mm w=0.18 mm a=5 mm E H (a) (b) (c) Fig. 5. 共a兲Multilayer structure of the fabricated prism of left-handed material. 共b兲Details of the structure of a layer consisting of a printed circuit board with resonant rings between a pair of foam slabs; the layer is covered by metal plates. 共c兲Geometrical details of the resonant rings. Mounting Base with Pyramidal Foam Absorber Prism of PVC Prism of Negative-refractive Index 10.5 GHz Microwave Horn Fig. 6. Photograph of the experimental setup. A prism of left-handed material is placed in a mounting base covered with pyramidal absorbing foam. The mounting base is placed in the center of a goniometer, which allows a microwave horn receiver to rotate. A microwave horn emitter is shown on the other side. Both the emitter and the receiver are commercial horns used for physics demonstrations. The photograph also shows an ordinary prism made of PVC. Fig. 7. Radiation pattern obtained with the setup shown in Fig. 6for the ordinary or right-handed prism of PVC and the left-handed prism. Main lobes of radiation are obtained at 38° for the right-handed prism and at ⫺15° for the left-handed prism. 351 351Am. J. Phys., Vol. 79, No. 4, April 2011 Velazquez-Ahumada et al. greatly reduced by the presence of the foam absorber, but the number of layers in the metamaterial was sufficient to obtain a measurable signal 共the width of the slot in the mounting base corresponds to the thickness of the multilayered structure of three flat unit cells, that is, 36 mm, which is also approximately one wavelength at 10.5 GHz兲. The results in Fig. 7for the radiation pattern of the PVC prism show a main lobe of transmission for a positive angle of ⬇40°. For an incidence angle of 20°, Eq. 共8兲leads to an index of refraction of n1=1.8. This result is very close to the expected value based on the relative permittivity of PVC,9which is close to n⬇冑3⬇1.7. When the metamaterial prism is measured in our setup, a main lobe is obtained for a negative angle of ⫺15° 共see Fig. 7兲. This negative angle is close to the value predicted by the calculated negative index of refraction of ⫺0.55. A minor lobe is also obtained when the transmitter and the receiver are aligned, corresponding to a positive angle of 20°. This minor lobe appears just in front of the emitter and can be explained by the fact that the Ewald-Oseen extinction theorem10 is not fully satisfied in our medium due to the medium’s discrete and finite nature. The main sources of errors in the experiments include misalignment of the mounting base with respect to the goniometer arm and tolerances in the fabrication process of the splitring resonator media and in the distance between the metallic plates. We observed a maximum misalignment of 2° in the location of the maxima, which still lets us distinguish between positive and negative refractions. These errors will vary depending on the ability of the students and the quality of the components. We used a photoetching technique with a minimum resolution of 20⫻10−6 m, which is sufficient for our purposes. We also found that the alignment of the metallic plates so that they are reasonably parallel is crucial for the success of the experiment. We repeated our experiments inside a small anechoic chamber without significant changes in the results, which shows that the partial shielding we used is sufficient for the experiment. ACKNOWLEDGMENTS This work was supported by the Spanish Ministerio de Ciencia e Innovación under Project Consolider-EMET CSD2008-00066. a兲Electronic mail: [email protected] 1R. Marques, F. Martin, and M. Sorolla, Metamaterials with Negative Parameters: Theory and Microwave Applications 共Wiley, Hoboken, NJ, 2008兲. 2V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ␧and ␮ ,” Sov. Phys. Usp. 10, 509–514 共1968兲. 3R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 共2001兲. 4D. M. Pozar, Microwave Engineering 共Addison-Wesley, New York, 1993兲. 5J. D. Baena, L. Jelinek, R. Marques, and M. G. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 共2005兲. 6The software can be freely downloaded from 具james.eii.us.es/ srrCalculator/典. 7J. D. Baena, J. Bonache, F. Martn, R. Marqus, F. Falcone, T. Lopetegi, M. A. G. Laso, J. Garca-Garca, I. Gil, M. Flores-Portillo, and M. Sorolla, “Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines,” IEEE Trans. Microwave Theory Tech. 53, 1451–1461 共2005兲. 8See supplementary material at 具http://www.doi.org/10.1119/1.3537122典 for an example of the measurement procedure. 9B. Riddle, J. Baker-Jarvis, and J. Krupka, “Complex permittivity measurements of common plastics over variable temperatures,” IEEE Trans. Microwave Theory Tech. 51, 727–733 共2003兲. 10 M. Born and E. Wolf, Principles of Optics, 7th ed. 共Cambridge U. P., Cambridge, 1999兲. 352 352Am. J. Phys., Vol. 79, No. 4, April 2011 Velazquez-Ahumada et al.