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

6. Resonant Function of Influence

[35] We estimate at resonance frequencies the value of the function of influence in current (12), which at the resonant condition (15) (taking into account small thermal losses in the plasma) in a main (zero) approximation over small parameter (electrical dimension of the spheroid) has the form

(19)

where e_Re= 1- w_p²/w², e_Im= n_p (w_p³/w³), and n_p= n/w_p. The analytical dependence of (19) on the spheroid shape Y and its filling in by the vacuum a is rather complicated because of the presence of radicals in (17), except some limited region of the values of Y and a which we will consider below.

[36] The estimation of the resonant value of the function of influence at the extreme frequency (13) typical for an oblate spheroid gives the following dependence on the relative thickness of the plasma layer

(20)

With a decrease of the filling in of the oblate plasma spheroid by the vacuum cavity a, the value (20) increases considerably and exceeds the resonant coefficient of propagation for a sphere with a cavity [Bichoutskaia and Makarov, 2005]. Therefore the oblate shape of the plasma spheroid appears more preferable than the spherical one at the resonance in current.

Figure 4

[37] One can see in more detail the dependence of the resonant function of influence in current (19) on the spheroid shape b/a and its filling in a by vacuum in Figure 4a, where the function |K_I| is shown as a function of the spheroid shape by solid and dashed curves for the low-frequency and high-frequency branches at two values of the second layer thickness: a=0.2 and 0.8. The function |K_I| in Figure 4a is normalized to the modulus of the resonant function of influence in current |K_I0| of the plasma sphere with radius a surrounding the electrical dipole at the resonant frequency

(21)

The value |K_I0| (21) exceeds the unity by 1 or 2 orders of magnitude depending on how small are the loss parameters n_p and a³_p. We assumed at calculations that the terms in the denominator of (21) are equal, i.e.,

[38] Because of the complexity of expression (13), one can obtain the explicit analytical dependence of the resonant function in voltage K_V on the problem parameters only at F=1 and F=0.

[39] In the case when a spheroidal coating is not strongly prolate and has a relatively thick first layer ( a₀³ 1, a₀³ Y, F approx 1 ), the resonant function |K_V| (13) appears close to the resonant function of influence in current |K_I|.

[40] In the case F=0 ( a₀=1, i.e., there is no first layer), the function K_V (13) is a resonance function of influence in voltage of a one-layer plasma coating of the spheroidal slot antenna [Bichoutskaia and Makarov, 2003]

(22)

The function decreases monotonously with a change of the spheroid shape Y from strongly prolate to strongly oblate one. This dependence of the function |K_V1| normalized to (21) is shown in Figure 4f for two values of the thickness of the plasma layer a₀=0.2 and 0.8.

[41] For the decreasing relative thickness of the first layer ( 0.2 a₀<1 ) the resonant function |K_V| (13) normalized to (21) versus the changing shape of the spheroid is shown in Figure 4b, Figure 4c, Figure 4d, and Figure 4e by solid and dashed curves for the low-frequency and high-frequency resonances, respectively, at two values of the relative thickness of the plasma layer a=0.2 and 0.8.

[42] For the relatively thick first layer with a₀ =0.2, the function |K_V| (curves 0.2 and 0.8 in Figure 4b) does not differ from the resonant function of influence in current |K_I| (curves 0.2 and 0.8 in Figure 4a), having for the oblate spheroid shape ( a approx 2b ) the maximum value exceeding by an order of magnitude the resonant function for sphere |K_I0|. With a decrease of the first layer thickness for a₀=0.8 and a₀=0.9 (curves 0.2 in Figures 4c and 4d) the extreme value of |K_V| (almost not changing in magnitude) shifts into the region of more prolate spheroid shape. In the case of the spherical shape of the plasma coating, the value of the resonant function of influence |K_V| may considerably exceed the value |K_I|, the latter statement following from the comparison at a=b of the values of these functions shown by curves 0.2 in Figures 4c and 4a. At further depletion of the thickness of the first layer ( a₀=0.9 and a₀=0.999 ), the high-frequency resonance |K_V⁽²⁾| (dashed curves in Figures 4d and 4e) is transformed into the resonance |K_V1| of a one-layer plasma coating (curves 0.2 and 0.8 in Figure 4f). The latter resonance has a maximum value at the strongly prolate spheroid shape, for which the resonant value of the field is not so high because of low values of the excitation coefficient D_0V.

[43] One can show that at a small relative thickness of the first layer dY 1, ( d =1- a₀³ ) for the branch of lower resonant frequencies in voltage (16)

the modulus of the function of influence (13)

decreases down to zero with a decrease of the thickness of the first layer. We will take the low-frequency resonance vanishing at the first layer thickness d₀ approx

(n_p²/y^*) (1- y^*)³, at which the value |K_V⁽¹⁾| becomes equal to unity.

[44] The modulus of the function of influence (13) |K_V⁽²⁾| for the branch of higher resonance frequencies (16) is close to the modulus of the resonance function of influence |K_V1| of a one-layer plasma coating (22). This is confirmed by the comparison of dashed curves in Figures 4e and 4f.

[45] Thus, with a decrease of the thickness of the first layer of the plasma coating, the low-frequency resonant function in voltage |K_V⁽¹⁾| approx 0 disappears, whereas the high-frequency function |K_V⁽²⁾| tends to |K_V1|. Therefore one can interpret the low-frequency resonance in voltage as a resonance of the plasma with the inner vacuum unlike the resonance in current [Bichoutskaia and Makarov, 2002, 2005].]