负折射率球散射的计算

Extinction properties of a sphere with negative permittivity and permeability R. Ruppin* Department of Physics and Applied Mathematics, Soreq NRC, Yavne 81800, Israel Received 17 August 2000; accepted 24 August 2000 by S. Ushioda Abstract Extinction spectra of spheres made of materials having dispersive permittivity and permeability are calculated. A dispersion in the microwave region, similar to that which has recently been achieved in artificial composite media is assumed. When either the permittivity or the permeability (but not both) is negative, there occurs a band of strong surface polariton absorption. However, when both quantities are negative, a band of highly suppressed extinction appears in the spectrum. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: D. Optical properties; E. Elastic light scattering PACS: 78.35.1c A medium for which both the dielectric constant e and the magnetic permeability m assume negative values will have peculiar properties. This has been demonstrated theoreti- cally by Veselago [1]. He has used the term “left-handed” for such a material, because for a plane wave propagating in such a medium, ~E £ ~H lies in the direction opposite to that of the wavevector. Other unusual electrodynamic properties of left-handed materials are reverse Doppler shift, inverse Snell effect and reverse Cerenkov radiation [1]. Recently, a material which is left-handed for frequencies in the micro- wave range, has been built by combining two dimensional arrays of split-ring resonators and wires [2]. It is expected that these techniques will be further developed, so that in the near future isotropic left-handed materials will also become available. Here we investigate the electromagnetic scatter- ing properties of a sphere made of a left-handed material. Simultaneous negative values of e and m can be realized only when there exists a frequency dispersion [1]. In our calculations we therefore employ dispersive forms of e and m , similar to those that can be attained artificially [2]. We consider a sphere of radius a, having relative permit- tivity e1…v† and relative permeability m1…v†; placed in a medium having real, frequency-independent permittivity e 2 and permeability m 2. The sphere is irradiated by a plane wave, and its scattering and absorption properties can be specified in terms of the Mie coefficients. For these we employ the most general form, which includes magnetic effects. In Stratton’s notation [3], they are given by an ˆ 2 m1jn…k1a†‰k2ajn…k2a†Š 0 2 m2jn…k2a†‰k1ajn…k1a†Š 0 m1jn…k1a†‰k2ahn…k2a†Š 0 2 m2hn…k2a†‰k1ajn…k1a†Š 0 …1† bn ˆ 2 e1jn…k1a†‰k2ajn…k2a†Š 0 2 e2jn…k2a†‰k1ajn…k1a†Š 0 e1jn…k1a†‰k2ahn…k2a†Š 0 2 e2hn…k2a†‰k1ajn…k1a†Š 0 …2† Here jn and hn are spherical Bessel and Hankel functions, respectively, and the primes denote differentiation with respect to their arguments. The propagation constants of the sphere and the surrounding medium are k1 ˆ e1…v†m1…v† p v=c and k2 ˆ e2m2p v=c; respectively. The extinction cross-section of the sphere, in units of the geometric cross-section, is given by Q ˆ 2 2…k2a†2 X1 nˆ1 …2n 1 1† Re…an 1 bn† …3† For the sphere dielectric constant we use the plasma-like Solid State Communications 116 (2000) 411–415 0038-1098/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0038-1098(00)00362-8 PERGAMON www.elsevier.com/locate/ssc * Fax: 1972-8943-4157. E-mail address: ruppin@ndc.soreq.gov.il (R. Ruppin). R. Ruppin / Solid State Communications 116 (2000) 411–415412 Fig. 1. Real part of the permittivity (lower curve) and the permeability (upper curve) used in the model calculations. The material is left-handed between 4 and 6 GHz. Fig. 2. Extinction cross-section of a sphere having a dielectric constant of the form (4), and m1 ˆ 1: The sphere radius is: (a) 2 cm; (b) 4 cm; (c) 8 cm. form e1…v† ˆ 1 2 v2p v…v 1 ig† …4† Such a dielectric constant, with the plasma frequency in the GHz range, can be realized by using a network of thin wires [4]. For the sphere permeability we employ the following form, which can be achieved by using a periodic arrange- ment of split ring resonators [5]: m1…v† ˆ 1 2 Fv 2 v2 2 v20 1 ivG …5† In the numerical calculations we use the value vp ˆ 10 GHz for the plasma frequency, and v0 ˆ 4 GHz for the magnetic resonance frequency. For the damping parameters the values g ˆ 0:03vp and G ˆ 0:03v0 are assumed. For the parameter F the value of 0.56 has been chosen, so that the band of negative permeability, which begins at v 0, will extend up to 6 GHz. The frequency dependence of the real part of e 1 and m 1 is shown in Fig. 1. In all calculations the values e2 ˆ 1 and m2 ˆ 1 are used for the surrounding medium. First, we consider the independent contributions of the dielectric dispersion and the magnetic dispersion to the extinction. These we obtain by performing extinction calculations, once with m1…v† ˆ 1 and e1…v† from Eq. (4), and once with e1…v† ˆ 1 and m1…v† from Eq. (5). Fig. 2 shows the extinction spectrum of spheres having a dielectric constant of the form (4), and no magnetic contri- bution. The real part of the dielectric constant is negative over the whole frequency region, and the spectrum consists of peaks which are due to surface plasmon-polaritons, as has been discussed previously in the case of small metallic spheres [6,7], where analogous peaks appear in the visible or ultraviolet range. The peaks are due to resonances of the Mie coefficients bl …l ˆ 1; 2;…†; and they are classified as surface mode peaks because the corresponding field ampli- tudes are highest at the surface of the sphere, and they decay towards its center. In the very small sphere limit (not shown here) only one peak, due to the l ˆ 1 mode, appears. With increasing sphere size, more and more modes, correspond- ing to higher l values, contribute to the extinction, and the spectrum becomes broader and flatter. Fig. 3 shows the extinction spectrum of spheres having a permeability of the form (5), and no dielectric contribution. The peaks, due to magnetic polaritons, are of two types. Those above v 0 (ˆ 4 GHz) in the region where Re…m1† , 0; are due to magnetic surface polaritons, resulting from resonances of the Mie coefficients al. The peaks below v 0, where Re…m1† . 0; are due to magnetic bulk polaritons. It can be seen that for small spheres, the surface mode R. Ruppin / Solid State Communications 116 (2000) 411–415 413 Fig. 3. Extinction cross-section of a sphere having a permeability of the form (5), and e1 ˆ 1: The sphere radius is: (a) 2 cm; (b) 4 cm; (c) 8 cm. absorption dominates, and with increasing sphere size the ratio of bulk to surface mode absorption increases. Next, we proceed to the case in which both electric and magnetic contributions are present, i.e. the sphere material is characterized by e1…v† and m1…v† given by Eqs. (4) and (5), respectively. The extinction spectrum of a sphere of radius 5 cm is shown in Fig. 4a. The pure plasmon extinction, calculated with m1 ˆ 1 (Fig. 4b), and the pure magnetic R. Ruppin / Solid State Communications 116 (2000) 411–415414 Fig. 4. Extinction cross-section of a sphere of radius 5 cm: (a) with permittivity and permeability given by Eqs. (4) and (5), respectively; (b) with permittivity given by Eq. (4) and m1 ˆ 1; (c) with permeability given by Eq. (5) and e1 ˆ 1: Fig. 5. Same as Fig. 4, but for a radius of 10 cm. extinction, calculated with e1 ˆ 1 (Fig. 4c), are also shown. Since the plasmon-like and the magnetic excitations are roughly independent, we could expect the two extinction mechanisms to reinforce each other. This is indeed the case in the low-frequency region below v 0. However, in the region just above v 0, in which the real parts of both e 1 and m 1 are negative, this does not hold, and instead the two mechanisms are strongly suppressed. Thus, above about 4 GHz a strong, broad depression appears in the extinction spectrum. This feature persists when the sphere size is increased, as demonstrated by Fig. 5, which was calculated for a sphere of radius 10 cm. The width of the band of reduced extinction varies with the width of the range of negative permittivity and permeability. In Fig. 6 the extinc- tion spectrum of a sphere of radius 7 cm is shown for two choices of F. The full curve was calculated with F ˆ 0:56; as before. The dashed curve was obtained by using the reduced value of F ˆ 0:21: In the latter case the range of negative permittivity and permeability extends only up to 4.5 GHz. This leads to a corresponding reduction of the width of the region of reduced extinction. Finally, we present the physical interpretation of this extinction suppression effect. In frequency regions where either Re‰e1…v†Š , 0 or Re‰m1…v†Š , 0; but not both, the electromagnetic radiation cannot propagate inside the sphere material. However, it can resonantly excite surface polaritons, for which the amplitude decays inside the sphere. When both Re‰e1…v†Š , 0 and Re‰m1…v†Š , 0; the medium becomes transparent to the radiation (except for the small intrinsic absorption), so that no surface modes exist. References [1] V.G. Veselago, Sov. Phys. Uspekhi 10 (1968) 509. [2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [4] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, J. Phys.: Condens. Matter 10 (1998) 4785. [5] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. [6] R. Ruppin, in: A.D. Boardman (Ed.), Electromagnetic Surface Modes, Wiley, New York, 1982, p. 345. [7] U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Springer, Berlin, 1995. R. Ruppin / Solid State Communications 116 (2000) 411–415 415 Fig. 6. Extinction cross-section of a sphere of radius 7 cm, having permittivity and permeability given by Eqs. (4) and (5), respectively. For the parameter F, the values of 0.56 (full curve) and 0.21 (dashed curve) have been used.