Electromagnetic field of the ground-based source surrounded...

2. Formulation of the Problem for the Dipole Antenna

Figure 1

[4] We consider first an equivalent problem for a prolate plasma spheroid with the large and small semiaxes a and b and the eccentricity e located in the medium with the relative dielectric permeability e₃. In the center of the spheroid an electrical dipole oriented along the rotation axis of the spheroid is located (Figure 1). We assume that the relative dielectric permeability of the cold isotropic plasma is

at the chosen time dependence of the form exp(- i wt), where w_p is the circular plasma frequency of electrons and n is the effective collision frequency.

[5] We assume that the depleted ion coating formed around the transmitter has a form of a spheroid with the semiaxes a₁ and b₁ and with the same eccentricity e. The spheroid is filled in by the medium with the relative dielectric permeability e₁. The distance between the focal points of the inner and outer spheroids will be designated 2d₁ and 2d, respectively. Below we will take the electrical dimensions of the spheroids to be small, i.e., ka 1 and k(|e|)^1/2a 1, where k is the wave number in the vacuum.

[6] We will create the solution of the Maxwell equations in the outer medium e₃ (Im e₃>0 ) satisfying the principle of radiation at the infinity using the system of spheroidal functions [Morse and Feshbah, 1953] determined in the prolate spheroidal coordinate system x, h, j (1 x< infty , -1 h 1 ) related to the outer spheroid

(1)

Here and below the values of the magnetic and electric fields are multiplied by the impedance of the free space

and by

respectively. The following designations are used in (1):

is the angular spheroidal function of the first kind and

is the function related to the radial spheroidal functions of the first kind

and second kind

by the formulae

here and

In the region of the depleted ion layer e₁, we will create the solution using the system of functions determined in the coordinate system related to the inner spheroid:

(2)

(3)

here

and

[7] In the region of the plasma layer with the relative dielectric permeability e, we will create the solution using two systems of functions determined in the coordinate systems related to both inner and outer spheroids:

(4)

(5)

here

and

[8] The following designations are used in expressions (1)-(5): A_n are the multipliers of excitation of the field of the electric dipole in the unlimited space with the relative dielectric permeability e₁, R_n¹ and D_n¹ are the reflection and transmission coefficients, respectively, for the boundary between the inner medium and the plasma coating, and R_n and D_n are the reflection and transmission coefficients, respectively, for the boundary between the plasma coating and the outer medium.

[9] The excitation coefficients A_n of the electromagnetic field of the source located in the center of the spheroid have the form

where x₀ is the radial coordinate of the source, I is the current at the transmitter entrance, and l is its effective length not exceeding the length of the rotation axis of the inner spheroid and satisfying the condition

According to Morse and Feshbah [1953], in the quasi-static approximation

one can choose among A_n the main coefficients corresponding to the index n =1. Therefore one can limit expansions (2)-(3) by one spheroidal wave with n =1.

[10] Writing the boundary conditions of continuity of the tangential components of the field on the surface of both spheroids and using for the spheroidal functions the theorem of addition [Ivanov, 1968] (which makes it possible to reexpand the solution of one system of functions in terms of another system of functions related to the outer spheroid and also to expand the angular spheroidal functions in the plasma medium on the angular functions of the inner or outer medium

we obtain an infinite system of algebraic equations. In the quasi-statistics condition, one can limit the angular function expansion with the accuracy of the terms of the order of

and

by the main first term of the expansion with the coefficient

Similar error is obtained using the first term in the expansion in the theorem of addition.

[11] Then confining ourselves by the main terms of the expansions we obtain the boundary conditions on the inner x=x₁ and outer x=x_a surfaces of the plasma spheroidal layer in the form of a truncated system of four algebraic equations

(6)

for n=1. From equation system (6), we find the coefficient D₁ of the electromagnetic field propagation into the outer medium. Before finding expression for D₁, we will formulate the second equivalent problem in which a slot spheroidal antenna is used as an emitter.