The Method of Auxiliary Sources for the Thin Films Superimposed on the Dielectric Surface

Transcription

1 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM The Method of Auxiliary Sources for the Th Films Superimposed on the Dielectric Surface I. M. Petoev, V. A. Tabatadze, R. S. Zaridze Tbilisi State University, Laboratory of Applied Electrodynamics, 3, Chavchavadze Ave. Tbilisi, Georgia Abstract The diffraction problem of the time harmonic electromagnetic wave on the certa metal-dielectric structure was solved by means of the Method of Auxiliary Sources. The structure represents right parallelepiped dielectric. A lattice of th perfectly conductg films is superimposed on the surface of this dielectric. Selectg the parameters of this structure, case of resonance between the lattice elements and the dielectric, some terestg properties of the complex materials wide range of the frequency are revealed. A user friendly computer program has been created for numerical experiments of the stated problem; it gives ability to change the structures parameters and the cident wave order to study the properties of these kds of structures.. INTRODUCTION In the modern micro and nano-electronics, the devices with th films on the dielectric have wide applications. Therefore, theoretical and numerical study of such structures is a very important problem. In this article, the Method of Auxiliary Sources (MAS) is used, order to study electromagnetic properties of such structures. It is known, that MAS is efficient for solution of the diffraction problem on the volumous objects. The application of this method deduced to the construction of two auxiliary surfaces side and side of this object. These surfaces repeat the form of the object and are shifted from the surfaces at the certa distance. The auxiliary sources located on them describe the field the

2 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE opposite parts of divide space. In case of the th conductg films at the first glimpse the ability of the MAS is restricted, because it is not clear how to distribute the auxiliary surfaces. Accordg to the proposed algorithm the given film should be contued all direction by imagary surface at a certa distance. The size of this imagary surface is one of the auxiliary parameters, which fluence the exactness of the calculation. From the both sides of the obtaed figure, up and down, there are constructed auxiliary surfaces. Field on top of the film is described by lower auxiliary surface and under the film- by the upper one. The unknown amplitudes of the auxiliary sources on these surfaces are determed by the boundary condition satisfaction, as on the film, also on the imagary surface. On the both sides of the film tangential component of the total electric film must be zero. The boundary conditions on the imagary surface are: the tangential component of the electric and magnetic field must be contuous. Results of successful application of described algorithms are provided []. This article is the contuation of the mentioned work. Here, stead of the film the free media, there is considered a system of such perfectly conducted films, which are superimposed on the surface of an ordary dielectric. The motivation this article is also the will to extend the MAS ability to the solution of such kd of problems efficiently. On the other hand, usg the right selection of the resonant parameters of the system, it is desirable to get the effects of multiple resonances []. We are lookg such resonance frequencies, when both: the film s sizes and the films lattice parameters total are resonant together with the dielectric s geometric parameters. Exactly case of such resonances occur the terestg electromagnetic properties of the study structure, which correspond to the properties of the composite (complex) materials [3, 4, 5]. The numerical calculations were provided usg the created user friendly program package. There were found resonant parameters of the system, when it has the properties of the negative refractive dex and chirality, the certa frequency range of the cident field. The results were obtaed by parallel validation of the boundary condition satisfaction.

3 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM. THEORETICAL PART.. Problem statement. The structure under study represents dielectric parallelepiped with permittivity, on the surface of which is superimposed the lattice of the defed size rectangle conductg films (figure ). The corners and the vertices of this dielectric are replaced by the cyldrical and spherical surfaces with small radius of curvature, order to avoid the occurrence of undesirable currents and fields with the sgularities on the surfaces. This structure is illumated by a harmonic electromagnetic wave with 3 i t e dependence, polarized to the parallel of the films surface. The electromagnetic wave source can be located side of the studied structure on certa distance or side dielectric. Our aim is to fd the diffraction field side and side of the structure as well as the field the far zone. The problem consists the numerical analysis of this scattered field and the character of its polarization, as well as to study the terestg composite materials properties ability of this structure on the resonant frequencies. time.. Solution of the problem. A general algorithm for the problem solution is provided below. This algorithm can be used for the more complicate forms of dielectric and the conductg films. Together to the diffraction problem solution it can be considered also as an antenna problem, when the source of the cident field is side of this structure and we are terested radiated pattern far field. The problem is to fd the total ner E r, H r and er E r, H r scattered fields. If the source of cident field is located side of the structure, then, H r H r H r E r E r E r Figure. Considered structure and cident field orientation c, c

4 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE, H r H r E r E r If this source is located side of the structure, then: Here., H r H r E r E r,, H r H r H r E r E r E r c E r, H r and E r, H. r are the fields scattered by the structure side and side, respectively. Outer and ner total fields should satisfy the boundary conditions on the dielectric S surface and also on the surface of each conductg plate. On the S surface, total field tangential component contuity is required. From the both side of the conductg films there is required the condition, the tangential component of the total field must be zero. As the problem is 3D, these conditions should be satisfied for any tangential component on the dielectric and conductg surfaces. If they are satisfied along two orthogonal tangential and vectors, then it will be c satisfied along any other tangential (figure )). So: E rs E rs, H rs H rs, E rs E rs, H rs H rs. () Along the conductor surfaces total electric field s tangential component must be zero on the both sides of the pieces: E r 0 E r 0, () E r 0 E r 0 Figure. The MAS application 4

5 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM H The application of MAS order to fd the fields E r, H r and E r, r is more details described the works [, 6]. The scheme of auxiliary surfaces construction and the distribution of the auxiliary sources on them are presented on the figure. So, solution of the problem reduced to the solution of the lear algebraic equations to the unknown amplitudes of auxiliary sources. 3. RESULTS OF THE NUMERICAL EXPERIMENTS 3..Evaluation of the solution accuracy. Before we get some numerical results it is necessary to be sure that the solution is accurate and we have to determe the error of numerical calculation. For this reason, as examples, we estimate the deviation from the boundary condition satisfaction between the collocation pots on the 3 sides of the considered structure, the dimensions of which the unit of cident wave length is the given case (figure 3). Another 3 sides of the object are symmetrical, with the same values. In this case, ner and er auxiliary surfaces are shifted from the object surface on the distance and the unit of the wave length. The provided figures corresponds respectively 6 and 36 collocation pots on the squared wave length. It can be clearly seen that the crease of the collocation pots number decreases the solution error. Durg numerical experiments and all calculations were done when the boundary conditions deviation was not exceed the 3%, which can be the maximum of the numerical error. a) b) Figure 3. The deviation from the boundary condition satisfaction for a) 6 and b) 36 collocation pots per wavelength squared 5

6 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE 3.. The structure with the small number of the films. An image of the considered structure is presented on the Fig. 4. This structure represents a dielectric object with the permittivity 4, on two opposite sides of which there are superimposed 3 square shape conducted films. The cident field propagates along direction, which makes angle 450 to the X and Y axes, polarized to the parallel of the films surfaces, as it is shown on Fig. 5. Calculations were made to study multiple resonances and to defe optimal parameters for complex materials properties of the structure. Figure 4. Considered structure Figure 5. The cident field orientation Figure 6. Scattered field far zone Figure 7. The analysis of the polarization The numerical calculations showed that case of the dielectric dimension abc and the size of the film 0.5 ( the units of wavelength), which corresponds to the resonant values, the pattern of the scattered field appeared an additional lobe, the direction of which corresponds to the negative refraction (figure 6). 6

7 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM Respectively, the given structure with such parameters is similar to the material with negative refractive dex. The analysis of the polarization properties of the scattered field showed that along two ma lobes it has elliptical polarization with the opposite rotations direction (Fig. 7), which corresponds to the chiral metamaterials. The last one is better seen on the animation. The near total field distribution is given on the Fig. 8. Figure 8. The near total field distribution Figure 9. The dependence of the irradiated power on the cident field frequency Figure 0. Scattered field far zone In order to get the resonance wider frequency range, the irradiated power dependence on the cident field s frequency some range, for three different value of the structure s width has been studied. As it is shown on the Fig. 9, case of the width equal to.9 a wider resonance is observed. At that time the negative refracted lobe creased, which corresponds to the crease of this effect Fig. 0. Near field distribution 7

8 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE two mutual perpendicular sections is shown on the Fig. a) and b). It can be see the resonant (standg) near field side the structure (Fig. b). a) b) Figure. Scattered field far zone: a) XOY, b) XOZ planes 3.3. The structure with the big number of the film elements. In order to crease the effect of negative refraction and chirality, the structure with bigger amount of the film elements, than the previous case has been considered. On the upper side, as on the lower one we have already 9 films with same sizes and same distance between them (Fig. ). The dimensions of the dielectric parallelepiped the units of wavelength are abc0.35, which yet doesn t correspond to the multiple resonances. Besides this, here we see small lobe of the negative refraction, also the rotation of the polarization is small. The correspondg distribution of the far field pattern and near field two mutual-perpendicular sections are presented on figures 3, and 4 a), b). Figure. Considered structure Figure 3. The scattered field far zone 8

9 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM a) b) Figure 4. Near field distribution: a) XOY and b) XOZ planes After that there was considered multiple resonant parameters of the structure: abc ( the units of wavelength). In this case, the lobe of negative refraction significantly creased. The correspondg far field pattern and the near field distribution are presented on the Fig. 5 and 6 Figure 5. The scattered field far zone Figure 6. Near scattered field distribution The animation of the near field shows, that next to the right side of the structure there is the standg wave formed, which takes place case of multiple resonance. 4. CONCLUSION The teraction of the electromagnetic wave with the dielectric structures, superimposed by the conductg th plane films, has been studied. Selectg the parameters of the given structures, case of the resonance between the lattice elements 9

10 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE and the dielectric, there were exposed terestg properties of the negative refraction and the chirality comparatively wide frequency range. The crease of the film object number creases the resonant effects, which has creased the observed complex materials properties. The numerical calculation was conducted usg computer simulation by means of specially created user friendly program package. The method of the problem s numerical solution used the MAS. ACKOWLEDGMENT The work is done with the fancial support of the Shota Rustaveli National Science Foundation the Scope of the grant 5/5. REFERENCES [] I. Petoev, V. Tabatadze, D. Kakulia, R. Zaridze, application of the Method of Auxiliary Sources for the th films and open surfaces Journal of Communications Technology and Electronics, Moscow, 04, accepted for publication. [] I. Petoev, V. Tabatadze, R. Zaridze, The Double and Triple Resonances Investigation for the Cassi Lattice to the Dielectric. Journal of Applied Electromagnetism (JAE), Vol.5, No., 03, pp [3] N. Engheta, R. W. Ziolkowski, Metamaterials, Physics and Engeerg Explorations. John Wiley & Sons, Inc., 006. [4] G. V. Eleftheriades, K. G. Balma, Negative-Refraction Metamaterials. Fundamental Prciples and Applications. Hoboken: John Wiley & Sons, Inc., 005. [5] I.V. Ldell, A.H. Sihvola, S.A. Tretyakov, A.J. Viitanen, Electromagnetic Waves Chiral and Bi-Isotropic Media. Boston: Artech House, 994. [6] I. Petoev, V. Tabatadze, R. Zaridze, The Method of Auxiliary Sources Applied to Problems of Electromagnetic Wave Diffraction by Certa Metal Dielectric Structures. Journal of Communications Technology and Electronics, Vol. 58, No. 5, 03, pp