The spectrum of electromagnetic radiation of a relativistic electron moving in the magnetic field oscillating in two mutually transversal spatial directions with two different frequencies is analytically investigated. The spectral properties of radiation in the planar biharmonic undulator are also discussed. The technique based on the associated Bessel functions was effectively applied in our calculations. The effect of the magnetic field and the undulator parameters on the radiation of the fundamental as well as low and high harmonics was elucidated. It is demonstrated that the biharmonic undulator can be exploited to regulate the emission of certain selected harmonics and hence contribute to the development of the efficient devices with high extraction and narrow spectrum.

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Abstract:We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power-exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.

The generation of harmonics in two-frequency undulator in a self-amplified spontaneous emission free electron laser (SASE FEL) is studied in order to produce Roentgen radiation in a relatively compact sized installation. The dynamics of SASE FEL is analyzed with the help of the phenomenological model to obtain the maximum of the X-ray high-harmonic power. The model accounts for the properties of the undulator magnetic field and of the electron beam and includes the major sources of losses, such as the electron energy spread, etc. It is compared and calibrated with the existing data on a FEL experiment. The advantages of the two-frequency undulator for Roentgen SASE FEL are demonstrated and the possibility to generate powerful mild Roentgen radiation at already ~25 m length is shown. The evolution of the bunching coefficients for high harmonics is studied together with the evolution of the FEL-induced energy spread. The linear and non-linear regimes are explored for common and for two-frequency undulators The usage of the two-frequency undulator for cascade SASE FEL with high X-ray harmonic power and high-harmonic bunching coefficients with low-induced energy spread is proposed.

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