Scattering of dipole field by perfectly conducting disk

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Scattering of dipole field by perfectly conducting disk

Vadim A. Kaloshin

Kirill K. Klionovski



Scattering of electromagnetic waves by a perfectly conducting disk was researched in [1-33]. In particular, scattering of a plane electromagnetic wave by a perfectly conducting disk was considered in [1-15]. Scattering of field of an electric and magnetic dipole located on axis of a disk was researched in [16-32]. A paper [33] describes a scattering field of an electric dipole located arbitrarily relative to a disk. A special case of a dipole mounted on a surface of a disk was presented in [16-25]. Asymptotic expressions of a scattering field of an electric dipole with perpendicular to disc orientation are available in [19, 25, 29]. Asymptotic formulas of a scattering field of a magnetic dipole with parallel orientation are available in [20]. A rigorous solution is available in a case of a perpendicular electric [16, 17] and parallel magnetic [17] dipole. This solution is represented as a series of spheroidal functions.

The statement of problem.

The first purpose of present paper is obtaining asymptotic expressions for a scattering pattern of an electric and magnetic arbitrarily oriented dipole located on an axis of a perfectly conducting disk. The second purpose of the paper is obtaining a numerical solution of this problem. The third one is to compare numerical results obtained using the asymptotic expressions and the numerical solution with numerical results from the papers [19, 20, 29].

We considered two cases - the dipole oriented parallel and normally to the disk (Fig. 1). It is enough to determine the field of the arbitrarily oriented dipole.

Fig. 1. The dipoles on axis of the perfectly conducting disk



The asymptotic expressions for scattering pattern derived separately near and far from the axis Z. We use the uniform asymptotic theory of diffraction [34, 35] to determine the scattering field far from the axis, and results of paper [36] near the axis. The results of paper [36] were obtained in the physical theory of diffraction approximation [3].

Fields of the dipoles in free space.

Consider a perpendicular magnetic dipole

(1)

and a parallel magnetic dipole

(2)

in free space where, x, y, z are Cartesian coordinates; z₀ and x₀ are the unit vectors in direction of the Z-axis and the X-axis, respectively; д(x) - is the Dirac delta function; m_z, m_x - are moments of the magnetic dipoles.

Azimuthal and meridional components of field of the magnetic dipoles in free space are

(3)

in a case of the perpendicular magnetic dipole, and



(4)

in a case of the parallel magnetic dipole. Here, л is wavelength; r, и, ц are spherical coordinates. Bottom indexes и and ц correspond to the meridional and azimuthal field component, respectively.

Consider a perpendicular electric dipole

(5)

and a parallel electric dipole

(6)

Here, p_z, p_x - are moments of the electric dipoles. Components of field of the perpendicular and parallel electric dipole in free space are respectively

(7)

(8)



The asymptotic expressions of field far from the axis.

Let's define the asymptotic expressions for meridional and azimuthal components of field of the electric and magnetic dipoles. The dipoles locate on the axis of the disk of radius R on distance h above the disk (Fig. 1). These expressions for the angles far from the Z-axis in the uniform asymptotic theory of diffraction approximation are

(9)

(10)

The meridional and azimuthal component in (9), (10) corresponds to the meridional and azimuthal component in (3), (4), (7) or (8) of analyzed dipole , , or , respectively. The variable for a perpendicular magnetic dipole or a parallel electric dipole. The variable for a parallel magnetic dipole or a perpendicular electric dipole. The variable for perpendicular magnetic and electric dipoles, and for parallel magnetic and electric dipoles. In (9), (10) is the Heaviside function. Function is first term of asymptotic representation of the Fresnel integral . The arguments of the functions and are

(11)

Formulas (9) and (10) take into account the first and the second order of diffraction. The last term in (9) describes the second order of diffraction and it is determined using the uniform geometrical theory of diffraction.

The asymptotic expressions of field near the axis.

Let's define asymptotic expressions near from the Z-axis. These expressions in the physical theory of diffraction approximation have the form

(12)

for the meridional component of field, and

(13)

for the azimuthal component of field. The scattering field in (12), (13) is determined as

(14)

for the perpendicular magnetic dipole,

(15)

for the parallel magnetic dipole,

(16)

for the perpendicular electric dipole, and

(17)



for the parallel electric dipole.

The meridional component and the azimuthal component in (12)-(17) correspond to the meridional and azimuthal component in (3), (4), (7) or (8) of analyzed dipole , , or , respectively. In (14)-(17) J_n(u) is the Bessel function of order n and argument u, and

(18)

Let's define the front-to-back ratio for the parallel electric and magnetic dipoles. We define asymptotic expressions of the front-to-back ratio in the physical theory of diffraction using the asymptotic expressions (12), (15) and (17) near the Z-axis:

(19)

for the parallel magnetic dipole, and

(20)

for the parallel electric dipole, where



The expressions (19), (20) take the next form when the radius of the disk tends to infinity

(21)

for the parallel magnetic dipole, and

(22)

for the parallel electric dipole. Expression (21) shows that the front-to-back ratio of the magnetic dipole oscillate relative a constant when the radius of the disk increase. This constant and amplitude of oscillation doesn't depend on the disk's radius. This dependence of can be explained by the fact that an amplitude of field of the magnetic dipole on an edge of the disk decreases proportionally R. At the same time, amplitude of scattering field by the edge increases proportionally R. As a result, the front-to-back ratio oscillates relative to a constant value.

Expression (22) shows that of the electric dipole is infinite when . In this case, amplitude of field of the dipole on the edge of the disk decreases proportionally , and increases in proportion to the disk's radius.

Numerical research of scattering by a disk.

The scattering by a disc was investigated by solving a singular integral equation of the first kind

(23)

numerically. Here, is the Green's tensor; is the unknown electric current on the disk; is electric field of a dipole in free space; is surface of the disk. We used the method of moments and an algorithm described in [37] to solve the integral equation (23). The unknown electric current shall be decomposed over basis of triangular elements:

(24)

Here, are unknown current's amplitudes; r₀ is the radial unit vector; ц₀ is the azimuthal unit vector. Basic functions with bases 2T_r and 2T_ц have the form

(25)

for perpendicular magnetic dipole, and

(26)

parallel electric and magnetic dipoles were considered. The method of moments reduces the problem of scattering to a system of linear algebraic equations. A matrix of the unknown current's amplitudes can be determined solving this system:

(27)

Elements of matrix of own and mutual impedances are calculated by numerical integration of the Green's tensor components and the basic functions:

(28)

A representation of components in spectral form was used during calculations of the own and "nearest" mutual impedances in the matrix . The basic function was decomposed into a two-dimensional Fourier integral over flat sheets of electric current. Integral over the surface Sґ in (28) was determined by integrating fields of the flat sheets with components . A representation in source-wise form for a ring of radial and azimuthal current in the spherical coordinates was used for purpose of reducing computation time in calculating the "distant" mutual impedances in the matrix . Elements of a matrix which describes an interaction of field of a dipole and the current of the disk have the following form:

(29)

The scattering field is determined by summing the field of dipole (3), (4), (7) or (8), and the field of the electric current of the disk. The field of the disk's electrical current is determined by numerical integration of fields of radial and azimuthal electrical current's rings. The integration was performed over radius of the rings with amplitude distribution (24).

The numerical results.

A pattern of the perpendicular magnetic dipole located on distance h=0,4л above the disk of radius R=2л and R=0.5л is shown in Fig. 2 and 3, respectively. Red curve plots were obtained using numerical solution of the integral equation. Blue curve plots were obtained using asymptotic equations of the field near the Z-axis (13), (14). Green curve plots were obtained using asymptotic equation of the field far from the Z-axis (10).

Figures 4-7 show patterns of the parallel magnetic dipole which mounts on the surface of the disk of radius R=2л and R=0.5л. Curves in Fig. 4, 6 show patterns in Е-plane .Curves in Fig. 5, 7 show patterns in Н-plane . Red curve plots were obtained using numerical solution of the integral equation. Blue curve plots were obtained using asymptotic equations of the field near the Z-axis (12), (15) in Fig. 4, 6, and (13), (15) in Fig. 5, 7. Green curve plots were obtained using asymptotic equation of the field far from the Z-axis (9) in Fig. 4, 6, and (10) in Fig. 5, 7. Violet and cyan curves plot were obtained using asymptotic equations [20] of the field far and near from the Z-axis, respectively.

Fig. 2. Gain of the perpendicular magnetic dipole above the disk of radius R=2л



Fig. 3. Gain of the perpendicular magnetic dipole above the disk of radius R=0.5л

Fig. 4. Gain of the parallel magnetic dipole which mounts on the surface of the disk of radius R=2л in Е-plane

Fig. 5. Gain of the parallel magnetic dipole which mounts on the surface of the disk of radius R=2л in Н-plane



Fig. 6. Gain of the parallel magnetic dipole which mounts on the surface of the disk of radius R=0.5л in Е-plane

Fig. 7. Gain of the parallel magnetic dipole which mounts on the surface of the disk of radius R=0.5л in Н-plane

Patterns for a case of the perpendicular electric dipole which mounts on the surface of the disk of radius R=2л and R=0.5л are shown in Fig. 8 and 9, respectively. Blue curve plots were obtained using asymptotic equations of the field near the Z-axis (12), (16). Green curve plots were obtained using asymptotic equation of the field far from the Z-axis (9). Violet and cyan curves plot were obtained using asymptotic equations [19] of the field far and near from the Z-axis, respectively.

Fig. 8. Gain of the perpendicular electric dipole which mounts on the surface of the disk of radius R=2л

Fig. 9. Gain of the perpendicular electric dipole which mounts on the surface of the disk of radius R=0.5л

Figures 10-13 show patterns for the case of the parallel electric dipole located on distance h=0,4л above the disk of radius R=2л and R=0.5л. Curves in Fig. 10, 12 show patterns in Е-plane . Curves in Fig. 11, 13 show patterns in Н-plane . Red curve plots were obtained using numerical solution of the integral equation. Blue curve plots were obtained using asymptotic equations of the field near the Z-axis (12), (17) in Fig. 10, 12, and (13), (17) in Fig. 11, 13. Green curve plots were obtained using asymptotic equation of the field far from the Z-axis (9) in Fig. 10, 12, and (10) in Fig. 11, 13.

Fig. 10. Gain of the parallel electric dipole above the disk of radius R=2л in Е-plane

Fig. 11. Gain of the parallel electric dipole above the disk of radius R=2л in Н-plane



Fig. 12. Gain of the parallel electric dipole above the disk of radius R=0.5л in Е-plane

Fig. 13. Gain of the parallel electric dipole above the disk of radius R=0.5л in Н-plane

Plots 14, 15 show dependence of the front-to-back ration of the parallel magnetic and electric dipoles on radius of the disk. The magnetic and electric dipoles locate on distances h=0,5л and h=0,75л above the disk, respectively. The disk's radius is presented in proportions of wavelength. Red curves in Fig. 14 and 15 were obtained using asymptotic equations (19) and (20), respectively. Blue curve in Fig. 14 plots was obtained using asymptotic equation (21). Green points in Fig. 14 and 15 t were obtained using numerical solution of the integral equation.



Fig. 14. The front-to-back ratio of the parallel magnetic dipole above the disk

Fig. 15. The front-to-back ratio of the parallel electric dipole above the disk

Let's discuss the calculation results in a case when radius of the disk is 2л. The asymptotic expressions of field near the axis describe radiation pattern with relative accuracy less than 1 dB for sector of angles and . The asymptotic expressions of field far from the axis describe radiation pattern with relative accuracy less than 1 dB for sector of angles , except the case of the parallel magnetic dipole in H-plane for angles . Curves of radiation pattern obtained using the asymptotic expressions near and far from the axis coincide for sectors of angles and . Sector of coinciding increases when the disk radius increases. The relative accuracy of obtained asymptotic expressions decreases to 4 dB for angles in a case of the disk's radius is 0.5л.

The relative accuracy of obtained asymptotic expressions of field near the axis is better than relative accuracy of the asymptotic expressions [20]. Analysis shows that asymptotic formulas obtained in [29] for radiation pattern of perpendicular electric dipole located on the disk are incorrect.

The asymptotic expressions of the front-to-back ratio of the parallel electric and magnetic dipoles allow determining this ratio with accuracy less than 0.3 dB for any radius of the disk.

Asymptotic expressions for calculating a scattering field of an arbitrarily oriented magnetic and electric dipole located on axis of a perfectly conducting disk were obtained. The asymptotic formulas make possible to calculate scattering field in full space. A relative accuracy of the asymptotic formulas is less than 1 dB when radius of the disk more then 2л. The asymptotic expressions for the front-to-back ratio have the accuracy less than 0.3 dB for any radius of the disk.



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