Energetic characteristics of radiation of oscillating...

2. Case of Cold Plasma

[5] Let an electric dipole in the "proper" reference system is characterized by the dipole moment density P'= p'd³(x',y',z'), where p'= p'₀ e'_z exp (-iw₀ t') and w'₀ is the source oscillation frequency in this reference system (the values related to the source "proper" reference system are marked by dashes). We will consider the dipole moving along the direction of its dipole moment with a velocity of v= v e_z (v>0). In this case, in the "laboratory" reference system, the source also has only an electric dipole moment equal to p= p'1-b², where b=v/c. The surrounding medium taken to be homogeneous, isotropic, and nonabsorbing has permittivity ( e ) and permeability ( m ) which depend on the wave frequency but are independent of the wave vector.

[6] In this paper we will analyze the radiation power S and its spectral density s(w) averaged over a period. The general expression for S [Ginzburg and Frank, 1947b] may be written in the following form:

(1)

(2)

where n²(w) = e(w) m(w), w₀= w₀'(1-b²)^1/2, and 1(x) is the unit function of Heaviside:

One can see that the s(w) value presenting the spectral density of the radiation energy differs from zero within the frequency range determined by the inequality

(3)

Further analysis of the energetic characteristics depends on the choice of the medium model. In publications [see, e.g., Frank, 1942; Ginzburg and Frank, 1947b; Tyukhtin, 2004a] the simplest case of a medium without dispersion is considered in a most detailed way. Not discussing this problem we come to the analysis of the energy loss in cold plasma characterized by the permittivity e=1-w²_p/w² (where w_p is the plasma frequency) and permeability m=1. In this situation the solution of inequality (3) determining the frequency range of the radiated waves takes the form

(4)

Radiation occurs only in the case when the values w_1,2 are real, i.e., w₀'>w_p (it is worth emphasizing that the source frequency in the laboratory frame of reference equal to w₀= w₀'(1-b²)^1/2 may be even lower than the plasma frequency). One can easily see that the width of the radiation spectrum increases with an increase of the source motion velocity and decreases with an increase of the plasma frequency.

[7] The spectral power of the radiation has the form

(5)

Figure 1

Figures 1a and 1b show the dependencies of the spectral power of the radiation (in the units p'²₀w'⁴₀/3c³ ) on the dimensionless frequency W equiv

w/w₀' for various value of b and W_p equiv

w_p/w₀'. It is worth noting that at not too large values of the plasma frequency, the character of spectral distributions is similar to the one taking place in vacuum. However, if W_p approx

1 (Figure 1b), the frequency distributions of the power take an interesting peculiarity: at rather high velocity, the radiation spectrum is entirely located at W >1, that is, in the region of the frequencies exceeding the proper frequency of the oscillator w'₀.

[8] The total radiation power of an electric dipole obtained after substitution of (5) into (1) and calculation of the corresponding integral is written in the following form:

(6)

where W_1,2= w_1,2/w'₀. One can show that this expression is a monotonously decreasing function of both the velocity of the dipole motion and plasma frequency. At low velocities the function coincides with the accuracy up to the value of the order of b² with the radiation power of a motionless source:

In the ultrarelativistic regime when 1-b² 1, one can obtain

Figure 2

[9] Figure 2 shows the dependencies of the radiation power of an electric dipole on the velocity of its motion at several values of the plasma frequency. One can see that this dependency is insignificant if the plasma frequency w_p is not too close to the proper frequency of the source w₀'. It is worth emphasizing that, in spite of this, the radiation spectrum undergoes quite significant reconstruction at changes in the velocity, this fact being mentioned above.

[10] Concluding this section, we make some notes concerning radiation of the moving longitudinal magnetic dipole in cold plasma (more details on this problem are given by Tyukhtin [2004a]). One can show that the total power of radiation of a magnetic oscillator is

where m'₀ is the amplitude value of the magnetic dipole moment in the reference system of the source. Thus S_m does not at all depend on the source motion velocity, in spite of the significant dependence of the spectral composition of the radiation. At equal dipole moments and proper frequencies, an electric dipole is more effective emitter than a magnetic one, because S_m S (the equality of the powers of this two sources is reached only in the limit b 1 ).