Electromagnetic scattering of Gaussian beam by two-dimensional targets

Authors


Abstract

[1] On the basis of the equivalence principle and reciprocity theorem, the multiple scattering up to Nth order by N parallel two-dimensional (2-D) targets arbitrarily located in a Gaussian beam is considered. The first-order solution can be obtained by calculating the scattered field from isolated targets when illuminated by a Gaussian beam. However, it is almost impossible to find an analytical solution for the higher-order scattered field if the 2-D targets are not circular cylinders because of the difficulty in formulating the couple scattered field. In order to overcome this problem, the composite scattering field is studied by employing the technique based on the reciprocity theorem and equivalence principle, and a line integral solution up to Nth order is obtained. In this calculation, only the previous-order scattered field from scatterers and the equivalent surface electric and/or magnetic current density induced by the incident beam are required. Using the approach proposed in this paper, the bistatic and the monostatic scattering fields of a Gaussian beam by parallel inhomogeneous plasma-coated conducting circular cylinders are calculated, and the dependences of attenuation of the scattering width on the thickness of the coated layer, electron number density, collision frequency, and radar frequency are discussed in detail.

1. Introduction

[2] It is well known that Gaussian beam can give a better simulation of radar beam than plane wave. Therefore, in recent years, the problem of scattering of a Gaussian beam by composite scattering models has been the subject of extensive investigations. Because of the simplicity of the geometry and the interest in practical applications, scattering of plane wave or Gaussian beam by isolated cylinders or spheres has been well studied both theoretically and experimentally [Wu and Guo, 1997, 1998; Zimmermann et al., 1995; Chen and Cheng, 1964; Wu and Wei, 1995; Gouesbet et al., 1990; Doicu and Wriedt, 1997; Wang, 1985]. However, when the composite scattering field from discrete random media is studied, the interactions of electromagnetic wave between different scatterers should be taken into account. Because of the important influence on total field, higher-order scattered field has attracted the attention of many researchers. Unfortunately, to obtain the solution up to higher order, it is necessary to treat the electromagnetic interaction between objects that is not only nonplane wave in character but have nonuniformities in amplitude and phase. For this type of the problem, the exact analytical solutions cannot be found except for a very small number of cases [Yokota et al., 1986; Elsherbeni and Hamid, 1987; Hongo, 1978; Elsherbeni et al., 1993]. To overcome these difficulties, a new technique based on the reciprocity theorem [Sarabandi and Polatin, 1994; Li et al., 1998; Chiu, 1998; Kong, 2000] is proposed by Sarabandi and Polatin [1994] to evaluate the composite scattered field from two adjacent targets and an approximate solution for the scattered field up to the second order is obtained. However, the initial formulae, which are used to evaluate the second-order scattered fields, take the form of volume integral. In our work, the technique proposed by Sarabandi is improved by introducing the surface equivalent electric current and surface equivalent magnetic current [Kong, 2000]. Then, the higher-order solutions are simplified from the volume integral form to the surface integral form (for three dimension scattering problems) or the line integral form (for two-dimensional scattering problems). Thus the difficulty in evaluating the higher-order scattered field is reduced. On the basis of the improved technique, an approximate solution up to Nth order for the scattered field of a Gaussian beam from an array of N parallel adjacent two-dimensional (2-D) targets is derived. In this calculation, only the previous-order scattered field of objects and the equivalent surface electric and/or magnetic current density induced by the incident beam are required.

[3] Because plasma can efficiently absorb electromagnetic waves, it can be used as microwave absorbers [Laroussi, 1993; Liu et al., 2002; Tang et al., 2003; Robert, 1990]. The absorption is mainly dependent on several parameters such as the electron number density, the radar frequency, the thickness of plasma layer, as well as the momentum transfer collision frequency, etc. Therefore, in section 3, the technique proposed in section 2 is applied to obtain an approximate analytical solution for composite scattering field from N inhomogeneous plasma-coated conducting cylinders when they are illuminated by a Gaussian beam. In section 4, the bistatic and the monostatic scattering are discussed and the results are compared with numerical computations based on the time domain integral equation method. The dependence of attenuation of the scattering width on the thickness of the coated layer, the electron number density, the collision frequency and the radar frequency is discussed in detail.

2. Scattering of a Gaussian Beam by Parallel Adjacent 2-D Cylindrical Objects

[4] When we are interested in a limited region of space, it is well known that all uninteresting regions outside this space can be replaced by using equivalent sources, which include equivalent electric current and/or magnetic current [Kong, 2000; Chang and Harrington, 1977]. In general, when the composite scattering field from infinite length cylindrical targets is studied, the field outside the scatterers is of interest. Thus the cylindrical targets can be replaced by employing equivalent surface electric current and/or surface magnetic current. In this section, on the basis of the equivalence principle and the reciprocity theorem, a solution for composite scattering of Gaussian beam by cylindrical targets is derived.

[5] As shown in Figure 1a, N infinite length cylindrical targets, which are randomly distributed and parallel with each other, are illuminated by a Gaussian beam. equation image1i and equation image1i denote the electric and magnetic field of the incident beam, respectively. Without loss of the generality, suppose the incident Gaussian beam would induce an equivalent electric current density equation imagen and an equivalent magnetic current density equation imagen on the surface of cylinder n in the absence of the other scatterers. First, considering the equivalent electric current density equation imagen as the primary source, the electric and the magnetic fields produced by equation imagen in the presence of the other N − 1 cylinders are denoted by equation imageJn and equation imageJn. On the other hand, considering the equivalent magnetic current density equation imagen as the excitation source, the electric and the magnetic fields produced by equation imagen in the presence of the other N − 1 cylinders are denoted by equation imageMn and equation imageMn, respectively.

Figure 1.

Geometry of the scattering problem.

[6] Now, let us consider another situation where the source equation imagen and equation imagen are removed and a line electric current source equation imagee = equation image (equation imageequation image0) and a line magnetic current source equation imagem = equation image (equation imageequation image0) are placed at the far-zone observation point P, as shown in Figure 1b. Here, the unit polarization vector equation image(equation image or equation image) and equation image(equation image or equation image) are related by equation image = equation image, where equation image denotes the unit vector of the propagation direction of the scattering field. In the presence of the cylinders except for the cylinder n, the electromagnetic fields produced by equation imagee and equation imagem are denoted by equation imagee, equation imagee and equation imagem, equation imagem, respectively. Here, it should be emphasized that the fields equation imagee, equation imagee, equation imagem and equation imagem contain not only the fields excited by the line sources but also their scattered fields from the cylinders except for the cylinder n.

[7] Application of the reaction theorem [Kong, 2000] over the entire 2-D space results in

equation image

where L represents a closed circularity at infinity. At an infinite distance away from the source, the following two relations exist

equation image
equation image

[8] Substituting equations (2) and (3) into equation (1), the integral over L on the right-hand side vanishes. On the basis of the equivalence principle and the extinction theorem [Kong, 2000], also the line integral over the surface of the N − 1 cylinders vanishes since equation image × equation image = equation image × equation image = 0. Thus equation (1) can be written as

equation image

[9] Suppose the infinite length cylinders are all perfectly electric conducting scatterers, as well known, the surface magnetic current density is equation imagen = 0 and, on the basis of the reciprocity theorem, the elementary magnetic current source equation imagem is equal to zero, too. Then, equation (4) can be reduced to the following expression:

equation image

[10] Suppose that N perfectly magnetic cylinders are considered, the surface electric current equation imagen = 0 and, on the basis of the reciprocity theorem, the elementary electric current source equation imagee is equal to zero, too. Then, equation (4) can be written as

equation image

[11] In the case that the N cylinders are all dielectric, using equations (4)–(6), the following three equations are obtained:

equation image
equation image
equation image

[12] Because equation imagee = equation image and equation imagem = equation image, equations (7), (8) and (9) can also be written as the following forms:

equation image
equation image
equation image

[13] Then, using equations (10) and (11), the first-order scattering field from cylinder n and the secondary scattered field, i.e., the rescattered field from the other N − 1 cylinders when illuminated by the first-order scattered field of cylinder n, can be evaluated.

[14] In equations (10) and (11), if the multiple scattered field up to N − 1 order of equation imageed and equation imagemd from the cylinders except for the cylinder n are all taken into account, we should note that equation imagee and equation imagem can be obtained as

equation image
equation image

[15] In equations (13) and (14), the multiple scattered fields of equation imageed and equation imagemd from the cylinder n are not taken into account. If these scattered fields are also considered, equations (13) and (14) should be rewritten as the following expressions, i.e.,

equation image
equation image

Substituting equations (15) and (16) into equations (10) and (11), the two following equations are obtained:

equation image
equation image

In equations (13)–(18), equation imageed is the direct electric field generated by equation imagee = equation image and equation imagemd is the direct magnetic field excited by equation imagem = equation image. equation image and equation image (i = 1, 2, 3⋯N) are the multiple scattered fields from the cylinders when illuminated by equation imageed and equation imagemd, respectively.

[16] In the preceding discussions, we should note that the cylinder n is an arbitrary one among the cylinders. Therefore, for the other cylinders, the scattered field can be calculated by using the similar way. Thus the composite scattered field of Gaussian beam by the cylinders is obtained as

equation image

In equation (19), equation imageJn can be evaluated by equation (17). Meanwhile, using the relation equation image, equation imageMn is obtained by solving equation (18).

3. Electromagnetic Scattering of Gaussian Beam by Circular Cylinders

[17] In the previous section, the expressions, which can be applied to evaluate the scattered field from any adjacent 2-D cylindrical objects with known geometries and dielectric properties, are derived. Using the new approach, in this section an approximate analytical solution is derived for N circular cylinders whose axes are all parallel to the x axis.

[18] As shown in Figure 2, the assumption is made that the cylinders lie in the far-field region of each other; that is, the following conditions should be satisfied:

equation image

where λ denotes the wavelength of the incident wave, equation image and equation image is the distance between cylinder centers of cylinder l1 and l2, equation image and equation image are the radius of cylinder l1 and l2, respectively. Suppose a Gaussian beam, which propagates along the positive z direction, is incident on the cylinders. As shown in Figure 2, the incident Gaussian beam has its focal point at original point and W0 denotes the beam waist radius. Neglecting the time factor exp (−iωt), the spatial distribution of the amplitude of the electric component equation image in the z = 0 plane is given by [Wu and Guo, 1998]

equation image

where E0 denotes the amplitude of the electric component at the center of the beam and, in the following discussions, E0 is set as unit. The polarization equation image may be chosen to be either equation image (TM) or equation image (TE). In the next step, we will derive the approximate analytical solutions of the scattered field by the cylinders.

Figure 2.

Configuration of the cylinders and the incident beam.

3.1. Solutions for the First-Order Scattered Field

[19] When the observation point equation image is in the far-field zone, the expressions of equation image and equation image along − equation image are given by [Wang, 1994]

equation image
equation image

where equation image denotes the propagation direction of the scattered field, k0 is the wave number of the incident wave in free space and characteristic impedance Z0 = 1/Y0.

[20] Invoking the equivalent principle, suppose the equivalent electric and the magnetic current density on the surface of cylinder n are denoted by equation image and equation image, respectively. Using equations (17) and (18), the first-order scattered field of cylinder n can be expressed as

equation image
equation image

[21] Substituting equations (22) and (23) into equations (24) and (25), the first-order scattering field from cylinder n when illuminated by the incident Gaussian beam is obtained by employing Stratton-Thu formulation [Kong, 2000]

equation image

[22] In equation (26) the relation equation image is used. equation image is the position vector of the cylinder center of cylinder n. equation image, the bistatic scattered electric field amplitude vector of cylinder n when it is illuminated by the Gaussian beam, can be expressed as

equation image

where

equation image

and the expressions for aJI and bJI are given by Wu and Guo [1998].

[23] Then, the first-order scattered field of the Gaussian beam by the cylinders can be written as

equation image

where equation image is expressed as equation (26).

3.2. Solutions for the Second-Order Scattered Field

[24] In this part, the solutions for the secondary scattered field are derived. In equations (17) and (18), the secondary scattered field of cylinder l1 when it is illuminated by the first-order scattered field from cylinder n can be written as

equation image
equation image

[25] Here, equation image is the scattered field of equation image by cylinder l1 and equation image is the scattered field of equation image from cylinder l1. Under the approximation that equation image and equation image are considered as plane wave propagating along the − equation image direction, equation image and equation image can be written as

equation image
equation image

where equation image is the position vector of cylinder l1, equation image and equation image denote the electric and magnetic field scattering amplitude vector of cylinder l1 when it is illuminated by a plane wave [Ruck, 1970]. ρ″ is the distance between the axis of cylinder l1 and the point at equation image, that is ρ″ = equation image and equation image = equation image. Then, the secondary scattered from cylinder l1 can be written as

equation image
equation image

[26] Keeping in mind the conditions on the dimensions of, and the distance between, the cylinders as specified in equations (20), (34) and (35) can be evaluated analytically. In equations (34) and (35), noting that equation image is the position of the point on the surface of cylinder n, then equation image; thus

equation image

[27] Under this approximation, in equations (34) and (35), equation image and equation image are not functions of the integration variables. Therefore these two equations can be rewritten as

equation image
equation image

Using equations (37) and (38), the secondary scattered field equation image from cylinder l1 is obtained as

equation image

where equation image is the scattered electric field amplitude vector of cylinder l1 when it is illuminated by a plane wave and equation image is the scattered electric field amplitude vector of cylinder n when the incident wave is a Gaussian beam. Meanwhile, in equation (39), the relation equation image and Stratton-Thu formulation are both used. Because the cylinder n is an arbitrary one among all the cylinders, the second-order scattered field of the Gaussian beam by all the cylinders can be written as

equation image

3.3. Solutions for the Third-Order and Higher-Order Scattered Fields

[28] In this section, the solutions for the third-order and the higher-order scattered fields are derived. First, we derive the solutions for the third-order scattered field.

[29] Suppose cylinder n, l1 and l2 are arbitrary three scatterers among the cylinders, cylinder n can be considered as one scatterer and the cylinder-cylinder pair, which is composed of cylinder l1 and l2, as the other one. Because the first-order and the secondary scattered field has been solved in sections 3.1 and 3.2, as shown in equations (17) and equation (18), this problem can be reduced to calculate the multiple scattered fields equation image and equation image. In equations (17) and (18), the third-order scattered field can be written as

equation image
equation image

[30] Suppose a line electric current source equation image = equation image and a line magnetic current source equation image = equation image are placed at the point of cylinder center of cylinder n, here, the unit polarization vector equation image and equation image are related by equation image, where equation image denotes the unit vector of the propagation direction of the excited field by equation image = equation image. Because the radial dimension of the cylinders is very small relative to the distance between the cylinders, the electric field produced by equation image and the magnetic field produced by equation image at the point of cylinder l2 can be written as

equation image
equation image

where equation image.

[31] Considering the conditions on the dimensions of, and the distance between, the cylinders as specified in equation (20), the scattered fields of equation image and equation image by cylinder l2 are obtained as

equation image
equation image

[32] Suppose the field equation image and equation image as the incident field illuminate on the cylinder l1, using a similar method as in section 2, the secondary scattered field of cylinder l2 can be written as

equation image
equation image

[33] Keeping in mind the conditions on the dimensions of, and the distance between, the cylinders as specified in equations (20), (47) and (48) can be evaluated analytically. In equations (47) and (48), noting that equation image is the position of the point on the surface of cylinder l1, then equation image; thus

equation image

[34] Under this approximation, in equations (47) and (48), equation image and equation image are not functions of the integration variables. Therefore these two equations can be rewritten as

equation image
equation image

Then, substituting equations (50) and (51) into equations (41) and (42), the following two equations are obtained:

equation image
equation image

[35] In equations (52) and (53), noting that equation imageequation image (equation imageequation image, equation imageequation image) · equation imageequation image (−equation images, equation imageequation image) = equation imageequation image (equation imageequation image, equation imageequation image) · equation imageequation image (−equation images, equation imageequation image), then, the third-order scattered field from cylinder l1 is obtained as

equation image

where equation image and equation image are the scattered electric field amplitude vectors of the cylinder l1 and l2 when illuminated by a plane wave and equation image is the scattered electric field amplitude vector of cylinder n when the incident wave is a Gaussian beam. Meanwhile, in equation (54), the relation equation image and Stratton-Thu formulation are used. Because the cylinder n is an arbitrary one among all the cylinders, the third-order scattered fields of the Gaussian beam by all the cylinders can be written as

equation image

[36] Using the similar way, the jth-order scattered field by all the cylinders can be written as

equation image

[37] From the preceding discussions, the scattered field up to Nth other can be easily obtained, and its solution is written as

equation image

The composite scattered field of a plane wave by cylinders is obtained by replacing the scattered electric field amplitude vector of cylinder n when the incident wave is a Gaussian beam with that when the incident wave is a plane wave in equations (29), (40), (55) and (56).

4. Numerical Results

[38] The numerical electromagnetic code (NEC), which is a computational package based on the time domain integral equation method, we have only provides for plane wave. Therefore, to check the validity of the present method, the scattering results of plane wave from plasma-coated conducting cylinders are compared with the data obtained by using NEC. In the following discussions, only the TM case is considered because of the limited length of this article.

[39] In Figure 3, the frequency of the incident plane wave is 5GHz, the collision frequency and the electron density of the plasma are Ve = 50 GHz and Ne0 = 5.0 × 1017 m−3, respectively. The radius of the cylinders is 0.005 m and the coating thickness is 0.0025 m. From Figure 3, one can see that the difference between the scattered field up to second order and that up to third order is very small. Therefore the third-order and the higher-order scattered fields can be neglected. Then, in the following discussions, only the scattered fields up to second order are considered.

Figure 3.

Scattering width of two plasma-coated conducting cylinders versus scattering angles.

[40] Figure 4 gives the comparisons between the results calculated by employing the method proposed in this paper and the numerical computations based on the time domain integral equation method when the incident field is a plane wave. In Figures 4a–4d, the cases of three, four, five and six cylinders are discussed, respectively. Here, the radii of the cylinders and the parameters of the coated plasma layers are the same as in Figure 3. From Figure 4, it is obvious that the second-order results for these four cases provide reasonable approximations, and the results are in very agreement with the TDIEM data over the angular range. Then, the validity of the present method has been proved.

Figure 4.

Comparisons of the results calculated by using our method and TDIEM data. (a) Three cylinders: equation image = (0 m, 0 m), equation image = (0.1 m, 0 m), and equation image = (0.05 m, 0.1 m). (b) Four cylinders: equation image = (0 m, 0 m) equation image = (0.1 m, 0 m), equation image = (0.05 m, 0.1 m), and equation image = (0.15 m, 0.1 m). (c) Five cylinders: equation image = (0 m, 0 m), equation image = (0.1 m, 0 m), equation image = (0.2 m, 0.1 m), equation image = (0.05 m, 0.1 m), and equation image = (0.15 m, 0.1 m). (d) Six cylinders: equation image = (0 m, 0 m), equation image = (0.1 m, 0 m), equation image = (0.2 m, 0.1 m), equation image = (0.05 m, 0.1 m), equation image = (0.15 m, 0.1 m), and equation image = (0.25 m, 0.1 m).

[41] In the following discussion, the problem of scattering of Gaussian beam from two inhomogeneous plasma-coated conducting cylinders is considered. To deal with this kind of problem, the inhomogeneous plasma coating can be divided into many uniform plasma layers. Suppose the electron density profile of the inhomogeneous plasma coating takes the form of parabola distribution along radial direction. The electron density of each layer can be expressed as Ne(i) = Ne0Ne0 [(rir0)/d]2, where Ne0 is the center electron density and Ne(i) denotes electron density of the ith layer. r0 and ri are the radii of the conducting cylinder and the ith layer, d denotes the thickness of the inhomogeneous plasma coating.

[42] Figure 5 gives the scattering angle patterns of the bistatic scattering width from six cylinders. In Figure 5, the radius of the cylinders is 0.1 m and the thickness of the coating layers is 0.05 m. The position vector of each cylinder is equation image = (0 m, 0 m), equation image = (0.5 m, 0 m), equation image = (1.0 m, 0 m), equation image = (0.25 m, 0.5 m), equation image = (0.75 m, 0.5 m) and equation image = (1.25 m, 0.5 m), respectively. The center electron density and the collision frequency of the plasma layer are Ne0 = 3.0 × 1017 m−3 and Ve = 50 GHz, respectively. From Figure 5, it is found that the scattering width of plasma-coated cylinders for Gaussian beam and the plane wave incidence has a similar form, but the amplitude of the pattern for the plane wave is larger than that for the Gaussian beam. From Figure 5, one can also see that, with the increase of the beam waist radius, the results for Gaussian beam will gradually approach to that of the plane wave case. Figure 6 gives the backscattering width versus the radar frequency for different center electron densities. In Figure 6, the beam waist radius is W0 = 0.5 m, and the other parameters are the same as in Figure 5. As shown in Figure 6, with the increase of the center electron density, the attenuation of the backscattering width is greater and the attenuation bandwidth is broadened. This phenomenon can be explained by the reason that higher electron density will cause more intense collision between electron and the other particles and then the absorption of the plasma layer becomes stronger. Because the cutoff frequency of the plasma layer increases as increasing center electron density, from Figure 6, it is also observed that the radar frequency, which corresponds to the minimum value of the backscattering width, is increscent with the increase of the center electron density.

Figure 5.

Bistatic scattering width of six plasma-coated conducting cylinders for different beam waist radii.

Figure 6.

Backscattering width versus radar frequency f for various center electron densities.

[43] In Figures 7 and 8, the parameters are the same as in Figure 5. From Figure 7, one can observe that with the increase of the thickness of the plasma layer, the attenuation of the backscattering width is stronger. However, for different thicknesses of the plasma layer, the collision frequency that corresponds to the minimum value of the backscattering width is a constant value, which is equal to the angle frequency of the incident beam. This is because that resonance absorption will occur when collision frequency of the plasma is equal to the angle frequency of the incident wave. Figure 8 gives the backscattering width versus collision frequency for various center electron densities. From Figure 8, it is obvious that the attenuation of the backscattering width is stronger if the electron density is higher. As shown in Figures 7 and 8, for a given radar beam, the resonance attenuation point only depends on the collision frequency of the plasma.

Figure 7.

Backscattering width versus collision frequency Ve for different thicknesses of the coated plasma layer.

Figure 8.

Backscattering width versus collision frequency Ve for different electron densities.

5. Conclusions

[44] On the basis of the equivalence principle and reciprocity theorem, a general technique has been developed for deriving the scattered field up to Nth order from parallel 2-D targets and a surface integral solution up to Nth order is obtained. In this solution only the lower-order scattered field of objects and the equivalent surface electric and/or magnetic current density induced by the incident beam are required. The formulation has been applied to obtain approximate analytical solutions for multiple scattered fields of a Gaussian beam by an array of plasma-coated circular cylinders. The validity of the present method was verified by comparison with the time domain integral equation computations. Meanwhile, the effects of the electron density, collision frequency and thickness of the plasma layer on the scattering width are discussed in detail and some significant conclusions are obtained.

Acknowledgments

[45] This research was supported by the National Natural Science Foundation of China (grant 60571058), the National Defense Foundation of China, and Graduate Innovation Fund, Xidian University.

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