Harmonic generation by ultrarelativistic electrons in a planar undulator and the emission-line broadening

Abstract

We present an analytical treatment of the undulator radiation (UR), accounting for major sources of the spectral-line broadening, for example, the beam energy spread, the emittance, and a constant magnetic field, additional to the ideal periodic fields. The examples of high harmonic generation in one-and two-frequency undulators are considered with account for homogeneous and inhomogeneous broadening in real devices. Qualitative estimations are given as well as the exact analytical expressions for the UR spectrum, and intensity are obtained and simplified in the case of high-energy electrons. The compensation of the divergency by the constant magnetic field is demonstrated in the analytical form. The application for free-electron lasers (FEL) is demonstrated and compared with the radiation of a FEL on the supposition of an ideal undulator.

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

“…They benefit from the use of special functions [38][39][40]. The mathematical instruments, used to solve DE, generally range from a variety of integral transforms [41,42] to expansion in a series of generalized orthogonal polynomials [43] with many variables and indices [44][45][46], which arise naturally in studies of physical problems, such as the radiation and dynamics of beams of charges [47][48][49][50][51][52][53][54], heat and mass transfer [55][56][57][58][59], etc. Moreover, exponential operators and matrices are currently used also for description of such nature fundamentals as neutrino and quarks in theoretical [60][61][62][63][64][65] and in experimental [66][67][68] frameworks.…”

Operational method of solution of linear non-integer ordinary and partial differential equations

“…In many cases, these solutions are best formulated in terms of special functions and orthogonal polynomials when used for relevant models of physical processes. Hyperbolic, elliptic Weierstrass and Jacobi type, generalized Airy and Bessel type functions are used (Vitanov et al 2015 ; Dattoli et al 2008 , 2009 ; Appèl and Kampé de Fériet 1926 ; Dattoli 2000 ; Dattoli et al 2005 ; Zhukovsky 2014 , 2015a , b , c , d ); expansion in series of Hermite and Laguerre polynomials (Appèl and Kampé de Fériet 1926 ) are employed. These polynomials possess generalized forms with many variables and indices (Dattoli 2000 ; Dattoli et al 2005 ).…”

Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations

“…The method of operational solution of DE demonstrated in [7][8][9][10] is applicable to a wide spectrum of physical problems, described by linear partial differential equations (PDE), such as propagation and radiation from charged particles [11][12][13][14][15][16][17][18][19], heat diffusion [20][21][22], including processes not described by Fourier law, and others [23][24][25]. In the context of the operational approach, the operational definitions for the polynomials through the operational exponent are very useful [26].The operational exponent is also applied when describing the fundamentals of structures in nature, including elementary particles and quarks [27][28][29]; such modern mathematical instruments are also used for the theoretical study of…”

“…Now, expanding the exponential in Equation (17) in series and equating the terms on the rightand left-hand sides of (17), we obtain the following definition for a n (x):…”