Evaluation of the Sacttering Matrix of Flat Dipoles Embedded in Multilayer Structures

Transcription

1 PIERS ONLINE, VOL. 4, NO. 5, Evaluation of the Sacttering Matrix of Flat Dipoles Embedded in Multilayer Structures S. J. S. Sant Anna 1, 2, J. C. da S. Lacava 2, and D. Fernandes 2 1 Instituto Nacional de Pesquisas Espaciais, Brazil 2 Instituto Tecnológico de Aeronáutica, Brazil Abstract The expression for the scattering matrix of a dipole embedded in a multilayer planar structure is derived analytically. The structure is excited by an elliptically polarized plane wave incident at an oblique angle. Calculation of the electromagnetic fields present in a four-layer structure is performed in the spectral domain. The scattering matrices of electric and magnetic flat dipoles printed on the interface between two confined layers are evaluated using two polarimetric features. 1. INTRODUCTION Interest in retrieving parameters from natural targets out of microwave remote sensing data has increased lately. Full polarimetric SAR data make the analysis feasible, especially when polarimetric decomposition techniques are utilized. These techniques are based on the characterization of radar scattering from complex targets in terms of a linear combination of scattering from simpler ones, constitute an effective tool for the analysis of polarimetric data. From this point of view, knowledge of the scattering matrix from simpler targets is crucial for extracting information gathered from polarimetric SAR data. Besides, the closer to the exact solution the determination of the scattering matrix is, the more precise the scattering analysis will be. From polarimetric SAR images and using the knowledge of scattering matrix it is possible, for instance, to develop mathematical models for natural targets and to derive several polarimetric target descriptors. It is also possible to improve the classification accuracy of land cover and use, to identify the scattering mechanisms intrinsic on an ensemble of pixels, and to build up a sensor calibration processes. In this context, the scattering matrix computation plays a central role in the polarimetric radar remote sensing imagery. Therefore, the aim of this work is to develop an analytical expression for the scattering matrix of flat dipoles embedded in a multilayer planar structure illuminated by an elliptically polarized plane wave at oblique incidence. The theory behind the computation of the electromagnetic fields present in the structure is briefly described in Section 2. In Section 3, the scattering matrices for an electric and a magnetic flat dipoles printed on the interface z = l 1 (see Fig. 1) are developed and compared. 2. THEORY The structure under investigation, as depicted in Fig. 1, is composed of four isotropic linear homogenous layers stacked up in the z direction, that is: two confined layers located between free space (the upper layer) and ground (the lower layer). The lower layer, of complex permittivity ε g and complex permeability µ g, occupies the negative-z region. Each of the confined layers is characterized by thickness l n, complex permittivity ε n and complex permeability µ n, where n = 1, 2. Perfect electric and magnetic conductors of infinitesimal thickness that act as scattering elements are printed on each layer interface. The planar interface z = l 1 + l 2 separates layer 2 from free space. The layers are assumed to be unbounded along the transversal x and y directions. The development is based on a global rectangular coordinate system located on top of the ground layer (interface z = 0) lying on the xy-plane. The electromagnetic fields in a multilayer structure are determined through the methodological approach described in [1]. According to this methodology the structure is treated as a boundary value problem. The analysis is carried out by employing the spectral domain full-wave technique. The induced electric and magnetic surface current densities on the conductors are the virtual sources of the scattered fields. Firstly, the wave equations for every layer are solved in the Fourier domain, which leads to a system of differential equations. Application of the proper electromagnetic boundary conditions at each interface yields a set of twelve equations with an equal numbers of

2 PIERS ONLINE, VOL. 4, NO. 5, Figure 1: Lateral view of a planar structure with two confined layers. unknowns. The analytical solution of this system leads to the spectral Green s functions in simple, closed form. Combination of the Green s functions and the transformed surface current densities allows the determination of the transformed fields at any point of the multilayer structure. Finally, the electromagnetic fields in spatial domain are obtained from the inverse double Fourier transform. The stationary phase method [2] is used to obtain asymptotic expressions for the far electromagnetic fields scattered by the multilayer structure. From these asymptotic expressions and the knowledge of the current densities, the scattering matrix elements are completely determined for any incidence and scattering direction. Using this method the far scattered electric field is expressed by: E 0 (r, θ, φ) = ik 0 e ik0r cot θ{ˆθe 0z (k xe, k ye ) 2π r ˆφη 0 h 0z (k xe, k ye )}, (1) in the spherical coordinate system, where bold face letters represent vectors, η 0 is the intrinsic impedance of free space, k xe = k 0 sin θ cos φ and k ye = k 0 sin θ sin φ are the stationary phase points, k 0 is the wave number of the exciting wave, r is the distance between the receiving antenna and the target, and e 0z and h 0z are the longitudinal components of the spectral electromagnetic fields. As a note, the current densities induced on the conductors could be determined by the moment method, or other techniques. A particular structure containing electric and magnetic dipoles printed only on the interface z = l 1 is analyzed next. The dipole dimension along the x-axis of a standard rectangular coordinate system is considered much larger than that along the y-axis, such that the y-component of the surface current density can be neglected. For this case, the longitudinal components e 0z and h 0z, on the stationary phase points, are: e 0z (k xe, k ye ) = 4ω2 e {k xe Φ e 0j 1xe (k xe, k ye ) + k ye Θ e 0m 1xe (k xe, k ye )}, (2) h 0z (k xe, k ye ) = 4ω2 m { k ye Φ h 0j 1xe (k xe, k ye ) + k xe Θ h 0m 1xe (k xe, k ye )}, (3) where j 1xe (k xe, k ye ) and m 1xe (k xe, k ye ) are respectively the transformed electric and magnetic surface current densities, and the following factors together with the spectral variables represent the spectral Green s functions of free space. Φ e 0 = ε 1 γ 1 γ 2 (ε 1 γ g cos α 1 + iε g γ 1 sin α 1 ), (4) Θ e 0 = ωε 1 ε 2 γ 2 (ε g γ 1 cos α 1 + iε 1 γ g sin α 1 ), (5) Φ h 0 = ωµ 1 µ 2 γ 2 (µ g γ 1 cos α 1 + iµ 1 γ g sin α 1 ), (6) Θ h 0 = µ 2 γ 1 γ 2 (µ 1 γ g cos α 1 + iµ g γ 1 sin α 1 ), (7)

3 PIERS ONLINE, VOL. 4, NO. 5, e = 4ω 3 e iα0 {ε 1 γ 2 (ε 2 γ 0 cos α + iε 0 γ 2 sin α)(ε g γ 1 cos α 1 + iε 1 γ g sin α 1 ) +ε 2 γ 1 (ε 0 γ 2 cos α + iε 2 γ 0 sin α)(ε 1 γ g cos α 1 + iε g γ 1 sin α 1 )}, (8) m = 4ω 3 e iα0 {µ 1 γ 2 (µ 2 γ 0 cos α + iµ 0 γ 2 sin α)(µ g γ 1 cos α 1 + iµ 1 γ g sin α 1 ) +µ 2 γ 1 (µ 0 γ 2 cos α + iµ 2 γ 0 sin α)(µ 1 γ g cos α 1 + iµ g γ 1 sin α 1 )}, (9) γ m = ( 1) τ ω 2 µ m ε m (k 2 xe + k 2 ye) Im{γ m } 0, (10) with m {0, 1, 2, g}, α 0 = γ 0 d 2, α 1 = γ 1 d 1, α 2 = γ 2 d 2, α 3 = γ 2 d 1 and α = α 2 α 3. The τ variable, which defines the wave propagation direction, can take values of either 1 or 2. For the free space τ = 1, i.e., a wave propagating in the positive z-direction. For the inner layers, τ takes both values, and for the ground layer τ = 2 to represent a wave propagating in the negative z-direction. 3. SCATTERING MATRICES The scattering matrix can be seen as a mathematical characterization of target scattering. This matrix relates the electric fields of the wave scattered by a target to the ones of the incident wave. It is the most important parameter in polarimetric SAR analysis, since it provides complete information about the scattering mechanism; all polarimetric features that describe the target scattering can be derived from it. In order to determine the scattering matrix of a target in a multilayer structure, the procedure defined in [3] will be followed. This procedure directly relates the elements of the scattering matrix to the components of the scattered electric field; a standard spherical coordinate system (r, θ, φ) is chosen to coincide with a coordinate system (k, v, h) defined in terms of horizontal and vertical linear polarization components, as stated in [4]. The incident wave sets up currents on the scattering element (dipole), which in turn re-radiates a scattered wave. The currents induced on the dipole depend on the incident wave polarization. Therefore, from the reciprocity theorem, relation (11) between the components of the transformed far electric field in free space due to each linear polarization e (h) 0z (k xe, k ye ) = η 0 h (v) 0z (k xe, k ye ), (11) can be obtained, where the superscripts (v, h) are associated to polarization of the incident wave. Consequently, for an electric planar dipole the induced currents due to horizontal and vertical linear polarizations are, then, related by j (h) 1xe (k xe, k ye ) = C 1 ( kye k xe This relation leads to the following scattering matrix [S ed ] = C 2 cos φ s where the constants C 1 and C 2 are given by: ) j (v) 1xe (k xe, k ye ). (12) (cos φs ) 2 /C 1 sin φ s cos φ s sin φ s cos φ s C 1 (sin φ s ) 2, (13) C 1 = eφ h 0 m Φ e η 0 and C 2 = i2ω2 k0 2η 0Φ h 0 cos θ s j (v) 1xe 0 π (k xe, k ye ). m Similarly for the magnetic dipole, the relation between the induced currents and its corresponding scattering matrix are expressed, respectively, by: ( ) m (h) 1xe (k kxe xe, k ye ) = C 3 m (v) 1xe (k xe, k ye ), (14) [S md ] = C 4 sin φ s k ye where the constants C 3 and C 4 are represented by: (sin φs ) 2 /C 3 sin φ s cos φ s sin φ s cos φ s C 3 (cos φ s ) 2, (15) C 3 = eθ h 0 m Θ e η 0 and C 4 = i2ω2 k0 2η 0Θ h 0 cos θ s m (v) 1xe 0 π (k xe, k ye ). m

4 PIERS ONLINE, VOL. 4, NO. 5, Note that the constants C 2 and C 4, which are directly related to the currents induced on the dipoles, affect equally all the elements of the scattering matrices. The ability to distinguish different scattering mechanisms improves the accuracy of polarimetric SAR image classification. Several polarimetric features aiming at the characterization of the scattering mechanism are presented in the literature. One of these is the α-angle, which is derived from the eigenvalue/eigenvector decomposition [5]. Its physical interpretation is associated to the type of scattering mechanism and it can represent a wide variety of scatters. The α-angle for the electric and magnetic dipoles is computed from (16). { } α = cos 1 S xx + S yy Sxx S, with [S] = xy. (16) Sxx + S yy 2 + S xx S yy S xy 2 S xy S yy Even though the dipoles are of a different electromagnetic nature the α-angle for both is 45, a typical value of dipole scatter. Another feature used for the comparison between the scattering matrices of the electric and magnetic dipoles is a similarity measurement developed in [6]. It is based on square of a correlation coefficient and it is also independent of the spans of the scattering matrices and of the target orientation. The similarity measure is defined as: ρ([s 1 ], [S 2 ]) = kh 1 k 2 2 k k 2 2, (17) 2 where k 1 and k 2 correspond respectively to the modified Pauli-scattering vectors of the scattering matrices [S 1 ] and [S 2 ], as defined in [5], the superscript H denotes complex conjugate transpose, and 2 denotes the Euclidean norm of the vector. In the specific case of the scattering matrices of the electric and magnetic dipoles { (C 1 ρ([s ed ], [S md ]) = C 3 1)(C 1 C 3 1)(sin φ s cos φ s ) 2 } 2 [ C 1 2 (sin φ s ) 2 + (cos φ s ) 2 ] [ C 3 2 (cos φ s ) 2 + (sin φ s ) 2. (18) ] It can be noted that this coefficient carries the information of the scattering mechanism taking into account the electromagnetic parameters of the layers and the type of dipole. In particular, when the effect of the layers is neglected (i.e., the four layers are considered to be free space) both scattering matrices are non-similar, since the constants C 1 and C 3, are reciprocal real values, which yields ρ([s ed ], [S md ]) = CONCLUSIONS The full-wave technique in the spectral domain was used to derive the electromagnetic fields scattered by a multilayer planar structure. In the analysis, the structure was excited by an elliptically polarized plane wave at oblique incidence angle. From these fields, analytical expressions for the scattering matrices of electric and magnetic flat dipoles embedded in the structure were derived. Two polarimetric features were used as metrics to compare both scattering matrices: the α-angle and the correlation coefficient. Based on the former it was not possible to differentiate between the two dipoles, despite their different electromagnetic nature. On the other hand, it was shown that the latter feature can be used as a discriminating measure between these two scattering mechanisms. ACKNOWLEDGMENT The authors are grateful to FINEP-CAPTAER project for the partial support. REFERENCES 1. Lacava, J. C. S., A. V. Proaño De la Torre, and L. Cividanes, A dynamic model for printed apertures in anisotropic stripline structures, IEEE Trans. Microwave Theory Tech., Vol. 50, No. 1, 22 26, Collin, R. E. and F. J. Zucker, Antenna Theory: Part 1, McGraw-Hill Book Company, New York, Sant Anna, S. J. S., J. C. S. Lacava, and D. Fernandes, Closed form expressions for scattering matrix of simple targets in multilayer structures, Proceedings of International Geoscience and Remote Sensing Symposium, , Barcelona, Spain, July 2007.

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