where f t is an arbitrary
where f(t′) is an arbitrary function.
By averaging the components (18) of the particle velocity, we obtain:
As expected, the speed of the particle in Eq. (26) corresponds to the quantity described by the formula (22). It follows from Eq. (26) that the average transverse components of the particle momentum equal zero. For the average value of the longitudinal component of the particle momentum, we obtain the expression:
The average Senexin B of the particle is determined by the formula:
It is obvious from this formula, taking into account the expression (14), that depends on the intensity of the wave, its initial phase and polarization, the frequency of the carrier wave ω, the modulation frequency ω′, the cyclotron frequency , and the initial velocity of the particle.
The case of arbitrary wave polarization for a particle initially at rest Let us discuss the case when the particle is initially at rest () and is located in (0, 0, z0). Let us express the constants from the formula (13), taking into account that
For a wave with arbitrary polarization we obtain the following equality : where ρ is the ellipticity parameter ( corresponds to the linear, and to the circular polarization, while in other cases, (0 ≤ ρ ≤ 1) corresponds to the elliptical polarization). From the expression (17) we obtain the value of h at the initial time:
Let us say that where δ is the frequency ratio between and ω, and . Since within this problem we examine the acceleration of the charged particle in a high-frequency laser field with a constant uniform magnetic field but the radiation reaction is not taken into account, the particle energy should become infinitely large, since for the cyclotron authoresonance condition is satisfied. However, an infinitely large energy is impossible in real conditions, so this case can be excluded from consideration. Substituting the ration (32) into the expression (31), we obtain that where and is the intensity of the elliptically polarized electromagnetic wave, while is the wavelength. By substituting the expressions (29)–(34) into the formula (28), we obtain the average energy of the particle initially at rest in the wave of arbitrary polarization:
Further averaged over the initial phase Φ, the energy of the charged particle in the field of a plane frequency-modulated electromagnetic wave and in the constant uniform magnetic field is given by where
The resulting formulae (28), (29), (33), (35) and (36) for the average kinetic energy of the particle contain an explicit dependence on the initial particle velocity, the amplitude of the electromagnetic wave, the frequency modulation index, the frequencies of the carrier wave and the modulation, the cyclotron frequency, the intensity and its polarization, which allows making practical calculations. When μ ≪ 1, N = 1, the formulae (28), (29), (33), (35) and (36) take the form obtained in Ref. .
Conclusions This article offers the exact analytic solution of the equations for the charged particle motion in the external field of a frequency-modulated electromagnetic wave and in the constant magnetic field. We have formulated the dependence of the charged particle velocity on the intensity of a plane frequency-modulated electromagnetic wave with arbitrary polarization. The velocity in question depends on the amplitude and the polarization parameter of the electromagnetic wave, on the carrier, the modulation and the cyclotron frequencies. In the frequency-modulated electromagnetic wave (7) the fields E and H are periodic with their average values equaling zero. It could be assumed that the frequency-modulated electromagnetic wave and the constant uniform filed have an alternating effect on the charged particle, and that the average deviation caused by this effect is also zero. However, this assumption is incorrect. In particular, the particle in the field of a plane frequency-modulated electromagnetic wave systematically drifts in the direction of the electromagnetic field propagation. This is confirmed by an analytical calculation of the velocity and the momentum components, as well as of the average kinetic energy of a particle. With an increase in field intensity, according to the formula (23), the frequency of the oscillatory motion of the particle, the modulation frequency and the cyclotron frequency tend to zero. It was shown that the motion of the particle averaged over the oscillation periods and is a superposition of the constant motion and the vibrational motion with the carrier frequency, the cyclotron frequency and an nth vibrational motion with the frequency . In the absence of the frequency modulation, all the formulae are transformed into the respective formulae given in Ref. . The solutions have been obtained in the explicit dependence on the initial data, the amplitude of the electromagnetic wave, the carrier wave frequency, the modulation frequency, the wave intensity and its polarization parameter, which allows using the obtained solutions in practical calculations as well as the drift direction of wave propagation. The values of the momentum and energy of the particle, averaged over the period of vibration, were calculated. The oscillation period of the particle differs from that of the field. As the field intensity is increased, the frequency of the oscillatory motion of the particle tends to zero according to (23). It was shown that motion of the particle is the superposition of motion at a constant velocity and vibrational motion with the frequency of the electromagnetic field, cyclotron frequency and the frequency modulation different from the field frequency. In the absence of the frequency modulation, all the formulae go to the appropriate formulae given in . The solutions obtained are presented in the explicit dependence on the initial data, the amplitude of the electromagnetic wave, the wave intensity and its polarization parameter that allows everyone to apply the solutions in practice. The practical value of the study we carried out is that the results obtained can be used for designing various relativistic electronics devices. Furthermore, our results may be of interest for astrophysical research, and may also be used for interpreting the experiments with plasma in an external frequency-modulated electromagnetic field in the presence of a uniform magnetic field.