Negative-Index Refraction in a Lamellar Composite with Alternating. Single Negative Layers

Transcription

1 Negative-Inex Refraction in a Lamellar Composite with Alternating Single Negative Layers Z. G. Dong, S. N. Zhu, an H. Liu National Laboratory of Soli State Microstructures, Nanjing University, Nanjing 0093, China Negative-inex refraction is achieve in a lamellar composite with epsilon-negative ENG) an mu-negative MNG) materials stace alternatively. Base on the ective meium approximation, simultaneously negative ective permittivity an permeability of such a lamellar composite are obtaine theoretically an further proven by full-wave simulations. Consequently, the famous left-hane metamaterial comprising split ring resonators an wires is interprete as an analogy of such an ENG-MNG lamellar composite. In aition, beyon the ective meium approximation, the propagating fiel squeeze near the ENG/MNG interface is emonstrate to be left-hane surface waves with bacwar phase velocity. PACS: 78.0.Ci, 4.0.Jb, 4.70.Qs, 73.0.Mf

2 I. INTRODUCTION Electromagnetic composite metamaterials with simultaneously negative electric permittivity an magnetic permeability, calle left-hane materials LHM), have been a subject of scientific interest over the past several years [-5]. So far, there are a few ways towars the realization of an LHM. For examples, the metallic metamaterial comprising perioically arrange split ring resonators SRR) an wires was experimentally confirme to be an LHM [3]. The possibility of forming an LHM in conucting magnetic granular composite incorporating conucting ferromagnetic inclusions into an insulating matrix) was theoretically investigate [4]. More recently, Liu et al. propose a piezoelectric-piezomagnetic multilayer an foun that simultaneously negative an woul be available in such a structure [5]. It is easy to unerstan that, since there is no naturally existing LHM, the straightforwar ieas towars the realization of an LHM are how to combine single negative materials, namely, epsilon-negative ENG) material where < 0, but > 0 ) or mu-negative MNG) material where < 0, but > 0 ), uner appropriate volume fraction to get an ective LHM [3], or alternatively, how to get negative in materials having negative an vice versa. It was reporte that paire structure by ENG an MNG materials can lea to interesting characteristic, such as resonance, tunneling an transparency [6]. Guie moe along the interface of a pairing ENG-MNG waveguie was also theoretically analyze [7]. An intriguing phenomenon from the alternating layers of ENG an MNG materials is that it can

3 respon ectively in many ways lie an LHM, such as, the negative ective inex of refraction an the bacwar phase irection, except that an o not generally exist [8]. In this paper, we consier such an ENG-MNG lamellar structure to investigate its negative-inex refraction by full-wave numerical simulations. It is istinct from Ref. 8 in that the electromagnetic waves in this paper are incient parallel to the laminate slices. In Sec. II, base on ective meium approximation i.e., the thicnesses of the slices are small enough comparing with the electromagnetic wavelength), negative-inex refraction an planar slab focusing with simultaneously negative an are emonstrate in the structure, an the SRR-wire metamaterials are interprete as an analogy of the lamellar composite. Section III presents that, beyon the ective meium approximation, the electromagnetic fiel squeeze near the ENG/MNG interface is in fact left-hane surface waves with negative-inex refraction as well as bacwar phase irection. Finally, conclusions are given in Sec. IV. II. EFFECTIVE MEDIUM APPROXIMATION A. Left-hane lamellar composite Consier a lamellar composite as shown in Fig.. Two infinite slices, one is an ENG layer 0, > 0, thicness ) an the other an MNG layer < 0, < 0, thicness ), are stace alternatively along the x-axis. Base on > the ective meium approximation i.e., both an are extremely smaller than the electromagnetic wavelength), the components of an can be given by [9-] 3

4 x y x y + =, ) + + = z =, ) =, 3) + = z =. 4) + ) ) Therefore, the lamellar structure is a uniaxially anisotropic composite i.e., it is isotropic in the plane perpenicular to the x-axis). For simplicity without loss of generality, suppose =. Then, = < 0, 5) x y = x z = < 0. 6) x y = x z Such a structure can thus be treate as an inefinite meium in which not all of the principle components of an have the same sign []. What is of consierable interest is that, as long as the electromagnetic wave is incient with proper polarization, an inefinite meium here, the ENG-MNG lamellar composite) may be ientical in refraction property with an isotropic LHM []. Generally speaing, the prerequisites are as follows: <, an < S-polarize waves), 7) >, an > P-polarize waves). 8) Full-wave numerical simulations are performe to emonstrate the above conclusions. For the sae of simulations, it is not restricte to particular materials. As for the nonmagnetic ENG layer, it can be a metal, metamaterial of intersecting wires, 4

5 or piezoelectric material, etc., while for the MNG layer; it can be an antiferromagnetic material, SRR metamaterial, or piezomagnetic material, etc.. Consier two slices of an ENG layer = -0.4, = 0.) an an MNG layer =, = -.) with thicnesses = = 0. mm at the given frequency of 50 GHz. Accoring to Eqs. )-4), one gets,, ) = -, 0.8, 0.8), 9) = x y z,, ) = 0.44, -, -). 0) = x y z On the one han, when the electromagnetic wave is s-polarize i.e., electric fiel in the x-irection), as far as the refraction property is concerne, it shoul be equivalent to an isotropic an homogenous LHM having = = -. As shown in Fig. a), such a lamellar composite in wege shape is use to simulate its refraction fiel istribution with appropriate bounary conitions the surrounings of the wege is vacuum). From the intuitive viewpoint, the wege is cut in 45 angle so that the transmitte beam shoul be bent in 90 angle with respect to the incient s-polarize beam [see Fig. a)]. Bacwar phase moving is also confirme when the s-polarize wave propagates in the wege, which gives a further evience of the negative-inex refraction in such an ENG-MNG lamellar composite. On the other han, if the electromagnetic wave is p-polarize i.e., magnetic fiel in the x-irection), it shoul act as an isotropic meium with n = 0.59 = 0.8, = 0.44 ); this is consistent with the numerical result shown in Fig. b). Aitionally, from the simulations, it is foun the impeance matching conition between the wege an vacuum is η =, which is satisfie in the s-polarize case of Fig. a). However, 5

6 for the p-polarize wave, there are obvious reflections ue to the impeance of the lamellar composite mismatching to the vacuum. For another example, replacing the slices with an ENG layer = -0.5, = 0.5) an an MNG layer =, = -), from which n = - an = 0.75 for s-polarize wave, the reflections at the η wege interfaces are obvious though = an = in this case not shown). Planar slab focusing is another interesting feature of a homogeneous LHM. Though the lamellar composite is ectively an inefinite meium [see Eqs. 9) an 0)], it can still focus s-polarize waves as if it were an isotropic LHM ue to its in-plane isotropic property y = z = -). In Fig. 3, a current line source excites a cylinrical s-polarize wave with a frequency of 50 GHz. As is expecte, there are two equiistant images reprouce, one is in the center of the lamellar composite, an the other outsie of it. Aitionally, strong surface waves are also observe along the interfaces between the lamellar composite an vacuum, which is ue to the existence of surface polaritons [3]. B. Depenence on thicness ratio For brevity we have confine the iscussion uner the circumstance of =. In fact, accoring to Eqs. )-4), it is practical to achieve a certain negative n, by tailoring the ratio of the thicnesses, without resorting to new materials having ifferent an. C. Interpretation of the SRR-wire metamaterial After the emonstration that an ENG-MNG lamellar composite can ectively 6

7 act as an LHM uner the ective meium approximation, we have some remars on the famous negative refraction experiment by an SRR-wire metamaterial in wege shape [3]. The original iea was to obtain a metamaterial with its negative of the wire array an negative of the SRR array, an it was suppose that electromagnetic interactions between the SRR an wire arrays were negligible. However, the interpretation was questione in Ref. 4, in which the main viewpoint is that an LHM can not be achieve by simply placing wires negative ) in a homogeneous host with negative. From the viewpoint of this wor, the SRR-wire metamaterial can be regare as an analogy of the ENG-MNG lamellar composite, with wire layers as the ENG material an SRR layers as the MNG material [5]. Two relations are given to the SRR-wire metamaterial, no matter it is one-imensional, two-imensional, or three-imensional: SRR SRR wire wire =, ) SRR SRR + + wire wire SRR wire SRR wire =, ) + + wire SRR ) where an are permittivity components in the irection parallel to the SRR wire wire array, while an are permeability components in the irection SRR wire perpenicular to the SRR-plane. an are the intervals of SRR an wire SRR layers, respectively. III. BEYOND EFFECTIVE MEDIUM APPROXIMATION In this section, we consier the ENG an MNG materials with thicnesses larger enough than the electromagnetic wavelengths such that the ective meium wire 7

8 approximation is not applicable. Because neither the ENG nor MNG material itself supports the propagation of electromagnetic waves, therefore, the transmitte fiel, if any, is in fact squeeze near the interface layers. The wavevecter components in the lamellar composite can be written as x z c ω = -, 3) x z c ω = -, 4) where the transversal wavevector components, an, are relate as follows [3]: x x = 0 + x x S-polarize waves), 5) = 0 + x x P-polarize waves). 6) Remin that the thicnesses of the slices are much larger than the wavelengths in this case. Consequently, we obtain z c ω - ) - = S-polarize waves), 7) z c ω - ) - = P-polarize waves). 8) Accoringly, ective inex of refraction is etermine by n - ) - = S-polarize waves), 9) n - ) - = P-polarize waves). 0) From Eqs. 9) an 0), one can see that the surface waves transmitte along the 8

9 ENG/MNG interface has an ective inex of refraction. It is interesting an worth of emphasis that the ective inex of refraction is negative an hence left-hane surface wave is obtaine though no an can be erive uner this configuration; this is confirme in the following simulations. Consier again the ENG layer = -0.4, = 0.) an MNG layer =, = -.) at frequency of 50 GHz, but now with thicnesses = = 3 mm larger thicnesses are less computationally icient). Firstly, for s-polarize waves, negative refraction an anti-parallel phase velocity in the wege are emonstrate in Fig. 4a). The negative inex from the simulation by using Snell s law is consistent with that in Eq. 9), from where one gets n = Figure 4b) shows the cross-sectional fiel istribution. It is foun that the electromagnetic fiel is squeeze near the ENG/MNG interface an egenerate much quicly in the MNG layer than that in the ENG layer. This is because the transversal fiels away from the interface are evanescent exponentially into the ENG an MNG materials with the factors of exp- x) x an exp[-x x - )], respectively [3], where x x = 0.4 from Eq. 5. Seconly, such an ENG/MNG interface can not support the propagation of p-polarize waves, since one gets < 0 from Eq. 0. n IV. SUMMARY AND CONCLUSION In conclusion, negative-inex refraction is emonstrate in an ENG-MNG lamellar composite with full-wave simulations on the scenes of the wege-base negative refraction an planar slab imaging. Firstly, an of such a lamellar structure can be obtaine base on the ective meium approximation, which hence 9

10 can be regare ectively as a left-hane composite. Seconly, SRR-wire metamaterials can be interprete as an analogy to this moel. Thirly, When the ENG an MNG materials are thicer enough than the electromagnetic wavelength; the propagation of the electromagnetic fiel is emonstrate numerically to be left-hane surface waves with negative ective inex of refraction as well as bacwar phase irection. ACKNOWLEDGMENTS This wor was supporte by the State Key Program for Basic Research of China Grant No. 004CB69003), an by the National Natural Science Founation of China uner Contract No

11 [] D. R. Smith, J. B. Penry, an M. C. K. Wiltshire, Science 305, ). [] J. B. Penry, an D. R. Smith, Phys. Toay 57, ). [3] R. A. Shelby, D. R. Smith, an S. Schultz, Science 9, 77 00). [4] S. T. Chui, an L. Hu, Phys. Rev. B 65, ). [5] H. Liu, S. N. Zhu, Y. Y. Zhu, Y. F. Chen, N. B. Ming, an X. Zhang, Appl. Phys. Lett. 86, ). [6] A. Alu, an N. Engheta, IEEE Trans. Antennas Propag. 5, ). [7] A. Alu, an N. Engheta, IEEE Trans. Microwave Theory Tech. 5, ). [8] D. R. Frein, an A. Ron, Appl. Phys. Lett. 8, ). [9] D. J. Bergman, Phys. Rev. Lett. 44, ). [0] D. E. Aspnes, Am. J. Phys. 50, ). [] H. X. Da, C. Xu, Z. Y. Li, J. Magn. Magn. Mater. 85, ). [] D. R. Smith, P. Kolino, an D. Schurig, J. Opt. Soc. Am. B, ). [3] R. Ruppin, Phys. Lett. A 77, 6 000). [4] A. L. Porovsy, an A. L. Efros, Phys. Rev. Lett. 89, ); 9, ). [5] It is intuitive to mae such an analogy by comparing the ENG-MNG lamellar composite with a one-imensional SRR-wire metamaterial, which was illustrate in a number of articles. For examples, see Z. G. Dong, S. N. Zhu, H. Liu, J. Zhu, an W. Cao, Phys. Rev. E 7, ); an S. A. Cummer, an B.-I. Popa, Appl. Phys. Lett. 85, ).

12 FIG.. Schematic iagram of a lamellar composite with ENG an MNG materials stace alternatively. Only two perios of the multilayer structure are shown. FIG.. Color online) The magnetic fiel magnitues of wege-base refractions at 50 GHz incience. The wege-shape lamellar composite is cut in 45 angle with thicnesses = = 0. mm. a) S-polarize wave; b) P-polarize wave. FIG. 3. Color online) The magnetic fiel magnitue of planar slab imaging excite by a current line source at frequency of 50 GHz. FIG. 4. Color online) The magnetic fiel magnitues of wege-base refraction at 50 GHz s-polarize incience. The wege-shape lamellar composite is cut in 45 angle with thicnesses = = 3 mm. a) Longituinal istribution; b) Transversal istribution only one interface was simulate for clarity).

13 Fig. Z. G. Dong 3

14 Fig. a) Z. G. Dong 4

15 Fig. b) Z. G. Dong 5

16 Fig. 3 Z. G. Dong 6

17 Fig. 4a) Z. G. Dong 7

18 Fig. 4b) Z. G. Dong 8