Electromagnetic Scattering from an Anisotropic Uniaxial-coated Conducting Sphere

Transcription

1 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August Electromagnetic Scattering from an Anisotropic Uniaxial-coated Conducting Sphere You-Lin Geng 1,2, Xin-Bao Wu 3, and Bo-Ran Guan 2 1 Xidian University, China 2 Hangzhou Dianzi University, China 3 Shanghai Research Institute of Microwave Technology, China Abstract The scattering fields from an anisotropic uniaxial-coated conducting sphere by a plane wave are derived. The electromagnetic fields in uniaxial anisotropic medium and free space can be expressed in terms of spherical vector wave functions in uniaxial anisotropic media and isotropic medium. Applying the boundary condition in the interface between the uniaxial anisotropic medium and free space, the surface of the conducting sphere, the expansion coefficients of electromagnetic fields in uniaxial anisotropic medium are obtained, and then the expansion coefficients of scattering fields and radar cross sections can be obtained. Numerical results between this method and Mie theory are in good agreement as we expect. some numerical results are given in this paper. Introduction In recent years, there has been a growing interest in interaction between electromagnetic fields and anisotropic media, mainly due to its many applications in the fields of antennas and microwave devices, etc. As this is an interesting subject of many potential applications, there have naturally been some existing work, for instance, the analysis of two-dimensional geometries 1,2 and three-dimensional geometries 3-8. In this paper, on the basis of electromagnetic fields in uniaxial anisotropic medium using spherical vector wave functions 6, electromagnetic fields in anisotropic uniaxial-coated conducting sphere are formulated and numerically studied in this paper. The present work in this paper serves as a further extension of the studies in 6, and the fields in free space can be deduced from the present results and then expressed in terms of spherical vector wave functions in isotropic medium 6,9. Applying the boundary conditions of electromagnetic fields on the interface between uniaxial anisotropic medium and free space and on the interface of conducting sphere, all the field expansion coefficients in uniaxial anisotropic medium and free space are derived. Some numerical results are also obtained using the formulas and presented herein. One special case is considered, where the results obtained using the present method and the Mie theory 11 are compared to each other and a good agreement is observed. Formulas Let us consider an anisotropic uniaxial-coated conducting sphere illuminated by an incident plane wave. As illuminated in Fig.1, the coated sphere with outer radius a 1 and inner radius a 2 is located at the coordinate origin. On the surface the inner conducting sphere, the uniaxial anisotropic medium with permittivity tensor (ǫ) and permeability tensor (µ) is coated with thickness d(= a 1 a 2 ). It is assumed that the incident wave propagates in the +ẑ direction, the incident electric field has unity of amplitude, and is polarized in the + x direction. In the following analysis, a time dependence of exp( iωt) is assumed for the electromagnetic field quantities, but is suppressed throughout the treatment. The electric field vector wave equation in such a source-free uniaxial anisotropic medium can be written in the following form 2,5,6: µ 1 E(r) ω 2 ǫ E(r) =. (1) Figure 1: Geometry of a plane wave scattered by an anisotropic uniaxial-coated conducting sphere. where E denotes the electric field, while ǫ and µ represents the permittivity tensor and the permeability tensor

2 44 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August of uniaxial anisotropic medium, the expression are 6,8 ǫ = ǫ t ǫ t, µ = µ t µ t. (2) ǫ z µ z Using Fourier transform 5,6, the expansion of plane wave factors in terms of spherical vector wave functions in isotropic medium 1, and the properties of spherical Bessel functions 9, the electromagnetic fields (designated by the subscript 1 ) in the uniaxial anisotropic medium can be obtained as follows: E 1 = H 1 = l=1 q=1 l=1 q=1 n n A e q(θ k )M (l) (r, k q ) + B e q(θ k )N (l) (r, k q ) + C e q(θ k )L (l) (r, k q ) Pn m (cosθ k)k 2 q sin θ kdθ k, (3a) A h q(θ k )M (l) (r, k q ) + B h q(θ k )N (l) (r, k q ) + C h q(θ k )L (l) (r, k q ) P m n (cosθ k)k 2 q sin θ kdθ k. (3b) where n and n are summed up both from to + while m is summed up from n to n, and r is pointing in the (θ, φ)-direction in the spherical coordinates. The coefficients, F q, (l) are unknown, as in 6. A p q(θ k ), Bq p (θ k), Cq p (θ k) (where p = e or h) and k q are functions of θ k and they have been derived in 6. The vector wave functions, M (l), N (l), L (l) are spherical vector wave functions and they are also shown in 5,6,9,1 M (l) =z(l) n (kr) im Pm n (cosθ) e imφ θ dp m n (cosθ) e imφ φ, (4a) sin θ dθ =n(n+1)z(l) N (l) n (kr) P m n kr (cosθ)eimφ r+ 1 d(rz (l) n (kr)) dp m n (cosθ) θ+im Pm n (cosθ) φ e imφ, (4b) kr dr dθ sin θ (l) } dz L (l) =k n (kr) P m n d(kr) (cosθ)eimφ r + z(l) n (kr) dp m n (cos θ) θ + im Pm n (cos θ) φ e imφ. (4c) kr dθ sinθ where z n (l) (where l = 1, 2, 3, and 4) denotes an appropriate kind of spherical Bessel functions, j n, y n, h (1) n, and h (2) n, respectively. The incident electromagnetic fields(designated by the superscript inc) can be expanded in an infinite series in isotropic spherical vector wave functions 6,9,1 a x M (1) (r, k ) + b x N (1) (r, k ) δ m,1 + δ m, 1, (5a) E inc = H inc = k iωµ a x N (1) (r, k ) + b x M (1) (r, k ) δ m,1 + δ m, 1. (5b) where the expansion coefficients are defined as: a x = b x = i n+1 2n + 1 2n(n + 1), m = 1, i n+12n + 1, m = 1; 2 i n+1 2n + 1 2n(n + 1), m = 1, i n+12n δ s,l =, m = 1; 1, s = l,, s l. (6a) (6b) (6c)

3 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August According to the radiation condition of an outgoing wave (attenuating to zero at infinity) and the asymptotic behavior of spherical Bessel functions, only h (1) n should be retained in the radial functions, therefore the expansion of scattered fields (designated by the superscript s) are E s = A s M (3) (r, k ) + B s N (3) (r, k ), (7a) H s = k A s iωµ N (3) (r, k ) + B s M (3) (r, k ). (7b) where the coefficients, A s and Bs (n varies from to + while m changes from n to n), are unknowns to be determined, M (l) (r, k ) and N (l) (r, k ) denote the spherical vector wave functions defined in Eqs.(4a) to (4c), and k = ω(ǫ µ ) 1/2 identifies the wave number of free space, respectively. Applying the boundary conditions at the surface of uniaxial anisotropic medium, for example, when r = a 2, the expansion coefficients of electromagnetic fields in uniaxial anisotropic medium can be obtained by the following equations: l=1 q=1 n = B e q 1 k q r l=1 q=1 n = d dr ( rz (l) ) + Cq e A e q z(l) n (k qa 2 )P m n (cosθ k)k 2 q sin θ kdθ k =, } Pn m r (cosθ k)k q 2 sin θ k dθ k =. r=a 2 z (l) (8a) (8b) and r = a 1 it can be obtained the following expression Q (l) l=1 q=1 n = R (l) l=1 q=1 n = where expansion coefficients a x and bx following expression Q (l) q = A e 1 q k r R (l) q = +Cq h z (l) r iωµ A h 1 d q k k r dr +Cq e z (l) qpn m (cosθ k)k 2 q sin θ k dθ k = δ m,1 + δ m, 1 a x i (k a 1 ) 2, (9a) qpn m (cosθ k)k 2 q sin θ k dθ k = δ m,1 + δ m, 1 b x i (k a 1 ) 2. (9b) can be expressed in Eqs.(6a) and (6b). Q(l) q and R q (l) have the d rh (1) n dr (k r) z n (l) (k qr) iωµ r } h (1) n (k r) ( rh (1) n (k r) h (1) n (k r) } r=a 1, ) k z (l) n (k qr) B h q B e q 1 d rz n (l) k q r dr (k qr) 1 d rz n (l) k q r dr (k qr) From the Eqs.(8a) to (9b), it shows that firstly, the unknown coefficients of electromagnetic fields in the uniaxial anisotropic medium can be obtained; secondly, the coefficients of scattered fields in region are calculated; and lastly, the far scattering field of electromagnetic fields from an anisotropic uniaxial-coated conducting sphere by a plane wave, and the radar cross section are thus obtained. r=a 1. (1a) (1b)

4 46 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August Numerical Results and Discussion In the last section, we have presented the necessary theoretical formulation of the electromagnetic fields of a plane wave scattered by an anisotropic uniaxial-coated conducting sphere. To gain more physics insight into the problem, we will provide in this section some numerical solutions to the problem of electromagnetic scattering by an anisotropic uniaxial-coated conducting sphere. Numerical computations have been performed by applying the theoretical formulae derived earlier in the previous sections. In order to check the accuracy of the newly obtained numerical results, we performed one trial, that is, we calculated the radar cross sections using the present method and the Mie theory in reference 11. The results are shown in Fig.2, where electric dimensions of outer and inner spherical surfaces are k a 1 = 2.1π and k a 2 = 2π, while the permittivity and permeability tensor elements are ǫ t = ǫ z = 2.5ǫ, µ t = µ z = 1.6µ, respectively, (where and subsequently, ǫ and µ stand for the free space permittivity and permeability, respectively).it must be noted that the incidence wave propagates in the negative z-direction in this figure. Figure 2: Radar cross sections (RCSs) versus scattering angle θ (in degrees): Results of this paper (solid curve) and of Mie theory(block square)(fig.7 in Reference 11). Figure 3: Radar cross sections (RCSs) versus scattering angle θ (in degrees) in the E-plane (solid curve) and in the H-plane (short dashed curve). Figure 4: Radar cross sections (RCSs) versus scattering angle θ (in degrees) in the E-plane (solid curve) and in the H-plane (short dashed curve). From Fig.2, it is seen apparently that the radar cross sections calculated by using the two methods (i.e., the present method in this paper and Mie theory) are in very good agreement in both the E- and H-planes, where the maximum number of n used in Eqs.(8a) to (9b) is only 1 to achieve the convergence. It partially verifies the correctness and applicability of our theory as well as the program codes. After this, we obtain some new results unavailable elsewhere in literature. Two examples are considered herein, and their radar cross sections are plotted in Figures 3 and 4. Fig.3 represents radar cross sections of an anisotropic uniaxial-coated conducting sphere of more general uniaxial medium, where the permittivity and permeability tensor elements are characterized by ǫ t = 2ǫ, ǫ z = 4ǫ, and µ t = µ z = µ, the electric size of the uniaxial anisotropic spherical shell is chosen as k a 1 = 3π and k a 2 = 2.5π. The maximum number n in Eqs.(8a) to (9b) to achieve a good convergence is found to be 16. To illustrate further applicability of the scattering solution for an electrically large sized anisotropic uniaxialcoated conducting sphere(for example, in its resonance region), the radar cross sections of a relatively large uniaxial anisotropic sphere with k a 1 = 5π and k a 2 = 4π, under the illumination by an incident plane wave, are obtained and depicted in both the E-plane and the H-plane in Fig.4. The permittivity and permeability tensor parameters used for this case are: ǫ t = (2 +.2i)ǫ, ǫ z = (4 +.4i)ǫ, and µ t = µ z = µ. As the electric dimension of the sphere is increased, the maximum number of n used in Eqs.(8a) to (9b) must be significantly increased to 24 to achieve the convergence.

5 Progress In Electromagnetics Research Symposium 25, Hangzhou, China, August Conclusion The spherical vector wave function expansion solution to the plane wave scattering by an anisotropic uniaxialcoated conducting sphere is obtained analytically in this paper. The solution has only one-dimensional integral which can be calculated easily. Numerical results are obtained using the present method and compared with Mie theory and a fairly good agreement is observed. It is shown that the obtained solution is stable even for almost isotropic scatterers, since the proposed solution is an analytical one of the uniaxial anisotropic media, and the result of the Mie theory is a special case of the present method. The general numerical results, including the lossy anisotropic uniaxial-coated conducting sphere and resonance region, are given and are found reducible to those of spacial cases. The present analysis are believed to be useful in antenna and satellite communication system designs. *This work is partially supported through No. Y14539 by the Natural Science of Zhejiang Province of China and a Research Grant No: by the National Natural Science Foundation of China (NSFC). REFERENCES 1. Graglia, R. D. and P. L. E. Uslenghi, Electromagnetic Scattering from Anisotropic Material Part I: General Theory, IEEE Trans. on Antennas and Propagation, Vol. AP-32, No. 8, , Wu, X. B. and K. Yasumoto, Three-dimensional Scattering by an Infinite Homogeneous Anisotropic Cylinder: an Analytical Solution, J. of Appl. Phy., Vol. 82, No. 1, , Varadan, V. V., A. Lakhtakia and V. K. Varadran, Scattering by Three-dimensional Anisotropic Scatterers, IEEE Trans. Antennas and Propagation, Vol. AP-37, 8-82, Papadakis, S. N., N. K. Uzunoglu and C. N. Capsalis, Scattering of a Plane Wave by a General Anisotropic Dielectric Ellipsoid, J. Opt. Soc. Am. A, Vol. 7, No. 6, , Ren, W., Contributions to the Electromagnetic Wave Theory of Bounded Homogeneous Anisotropic Media, Phys. Rev. E, Vol. 47, , Geng, Y. L., X. B. Wu, L. W. Li and B. R. Guan Mie Scattering by a Uniaxial Anistropic Sphere, Phys. Rev. E, Vol. 7, No. 5, 5669/1-8, Tarento, R. J., K.-H. Bennemann, P. Joyes and J. Van de Walle, Mie Scattering of Magnetic Spheres, Phy. Rev. E, Vol. 69, 2666/1-5, Wong, K.-L. and H.-T. Chen, Electromagnetic Scattering by a Uniaxially Anisotropic Sphere, IEE Pt-H, Vol. 139, No. 4, , Wu, Z. S. and Y. P. Wang, Electromagnetic Scattering for Multilayered Sphere: Recursive Algorithms, Radio Sci., Vol. 26, No. 6, , Sarkar, D. and N. J. Halas, General Vector Basis Function Solution of Maxwell s Equations, Phy. Rev. E, Vol. 56, , Richmond, J. H., Scattering by a Ferrite-coated Conducting Sphere, IEEE Trans. Antennas and Propagation, Vol AP-35, No. 1, 73-79, 1987.