Free Electron Laser Equations Research Articles

Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation

Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation

Generalized Shehu Transform to $\Psi$-Hilfer-Prabhakar Fractional Derivative and its Regularized Version

In this manuscript, athours interested on the generalized Shehu transform of $\Psi$-Riemann-Liouville, $\Psi$-Caputo, $\Psi$-Hilfer fractional derivatives. Moreover, $\Psi$-Prabhakar, $\Psi$-Hilfer-Prabhakar fractional derivatives and its regularized version presented in terms of the $\Psi$-Mittag-Leffler type function. 
 They are also utilised to solve several Cauchy type problems involving $\Psi$-Hilfer-Prabhakar fractional derivatives and their regularised form, such as the space-time fractional advection-dispersion equation and the generalized fractional free-electron laser (FEL) equation.

A Note on Natural Transformation & Meijer’s G-Function

The objective of this paper is to investigate natural transform of Meijer’s G-Function in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties in terms of Mittag-Leffler functions. Further, we show some applications of these natural transform in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

Fox H-Functions in Self-Consistent Description of a Free-Electron Laser

A fractional calculus concept is considered in the framework of a Volterra type integro-differential equation, which is employed for the self-consistent description of the high-gain free-electron laser (FEL). It is shown that the Fox H-function is the Laplace image of the kernel of the integro-differential equation, which is also known as a fractional FEL equation with Caputo–Fabrizio type fractional derivative. Asymptotic solutions of the equation are analyzed as well.

Study of S-Function in Relationship with Natural Transform

The objective of this paper is to investigate natural transform of the S -function defined and studied by Saxena and Daiya [4] in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties in terms of Mittag-Leffler functions. Further, we show some applications of these natural transform in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

Shehu Transform of Hilfer-Prabhakar Fractional Derivatives and Applications on some Cauchy Type Problems

In this paper, we are interested on the Shehu transform of both Prabhakar and Hilfer–Prabhakar fractional derivative and its regularized version. These results are presented in terms of Mittag-Leffler type function and also utilized to obtain the solutions of some Cauchy type problems, such as Space-time Fractional Advection-Dispersion equation and Generalized fractional Free Electron Laser (FEL) equation, at which Hilfer-Prabhakar fractional derivative of fractional order and its regularized version are involved.

On the (k,s)-Hilfer-Prabhakar Fractional Derivative With Applications to Mathematical Physics

In this paper, we introduce the (k; s)-Hilfer Prabhakar fractional derivative. We discuss its properties and find a generalized Laplace transform of this operator. Furthermore, we show the applications of the (k; s)-Hilfer Prabhakar derivative in mathematical physics such as heat equation, free-electron laser equation, and kinetic differ-integral equation.

On certain fractional calculus operators and applications in mathematical physics

In this paper, we modify the (k, s) fractional integral operator involving k-Mittag-Leffler function and discuss its properties. We originate a new fractional operator named (k, s)-Prabhakar derivative and obtained some classical fractional operators as a special case of the newly proposed derivative. Some properties of the introduced operator are also part of the present work. The generalized Laplace transform is employed to study the characteristics of fractional operators. We modeled the free-electron laser (FEL) equation by involving the proposed derivative and can find the solution by using the said Laplace transform.

On the Elzaki transform and its applications in fractional free electron laser equation

Abstract This article is devoted to study Elzaki transform and its applications in Free Electron Laser equation involving Hilfer-Prabhakar fractional derivative. We derive formula of Elzaki transform for Hilfer–Prabhakar derivative and its regularized version. The solution of Free Electron Laser equation involving Hilfer-Prabhakar fractional derivative of fractional order is presented in terms of Mittag-Leffler type function. Furthermore, we find the application of the generalized Hilfer-Prabhakar derivative in linear partial differential equation and some problems of Mathematical Physics.

Fractional Derivatives, Memory Kernels and Solution of a Free Electron Laser Volterra Type Equation

The high gain free electron laser (FEL) equation is a Volterra type integro-differential equation amenable for analytical solutions in a limited number of cases. In this note, a novel technique, based on an expansion employing a family of two variable Hermite polynomials, is shown to provide straightforward analytical solutions for cases hardly solvable with conventional means. The possibility of extending the method by the use of expansion using different polynomials (two variable Legendre like) expansion is also discussed.

Hilfer–Prabhakar derivatives and some applications

We present a generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Furthermore, we show some applications of these generalized Hilfer–Prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations, and in difference–differential equations governing the dynamics of generalized renewal stochastic processes.

One-dimensional free-electron laser equations without the slowly varying envelope approximation

A set of one-dimensional equations has been deduced in the time domain from the Maxwell-Lorentz system with the aim of describing the free-electron laser radiation without using the slowly varying envelope approximation (SVEA). These equations are valid even in the case of arbitrarily short electron bunches and of current distributions with ripples on the scale of or shorter than the wavelength. Numerical examples are presented, showing that for long homogeneous bunches the new set of equations gives results in agreement with the SVEA free-electron laser theory and that the use of short or prebunched electron beams leads to a decrease of the emission lethargy. Furthermore, we demonstrate that in all cases in which the backward low frequency wave has negligible effects, these equations can be reduced to a form similar to the usual 1D SVEA equations but with a different definition of the bunching term.

Three-dimensional analysis of free-electron laser performance using brightness scaled variables

A three-dimensional analysis of radiation generation in a free-electron laser (FEL) is performed in the small signal regime. The analysis includes beam conditioning, harmonic generation, flat beams, and a new scaling of the FEL equations using the six-dimensional beam brightness. The six-dimensional beam brightness is an invariant under Liouvillian flow; therefore, any nondissipative manipulation of the phase space, performed, for example, in order to optimize FEL performance, must conserve this brightness. This scaling is more natural than the commonly used scaling with the one-dimensional growth rate. The brightness-scaled equations allow for the succinct characterization of the optimal FEL performance under various additional constraints. The analysis allows for the simple evaluation of gain enhancement schemes based on beam phase space manipulations such as emittance exchange and conditioning. An example comparing the gain in the first and third harmonics of round or flat and conditioned or unconditioned beams is presented.

Short-pulse length effects of the seed laser in high-gain harmonic generation free-electron laser

In order to generate coherent hard x-ray free-electron laser ( FEL), cascading stages of high-gain harmonic generation ( HGHG) scheme usually employ seed pulse with longitudinal length down to tens of femtoseconds. Such a short pulse length ( SPL) seed laser, with broad bandwidth in the spectral domain, can no longer be characterized by monochromatic light in FEL equations. In this paper, a set of self-consistent, multifrequency FEL equations is derived to describe the interaction between the seed laser and the electrons in the modulator of HGHG. Moreover, the SPL effects of the seed laser on the electron beam's energy modulation, the output pulse length, the output peak power, and the output wavelength tuning in HGHG are theoretically and numerically investigated. Study demonstrates that the SPL seed laser seriously and significantly influences the HGHG performances, and these influences have to be taken into account in the design and optimization of the seeded FEL scheme.

Analytical description of free-electron-laser oscillations in a perfectly synchronized optical cavity

We present an analytical description of free-electron-laser (FEL) oscillations in a perfectly synchronized optical cavity by solving the one-dimensional FEL equations. It is shown that the radiation stored in the cavity eventually evolves into an intense few-cycle optical pulse in the high-gain and low-loss regime despite the lethargy effect. The evolution of the leading slope of the optical pulse, which is defined from the front edge toward the primary peak, is found to play an important role in generating the intense few-cycle optical pulse. The phase space evolution of electrons which interact with the leading slope is solved analytically in a perturbation method, leading to an analytical solution for the optical pulse evolution. The peak amplitude and the pulse length at saturation are found to scale with the electron beam density and optical cavity loss. Those scalings agree well with the intense few-cycle pulses recently observed in a high-power FEL.

Analytical solution for phase space evolution of electrons operating in a self-amplified spontaneous emission free electron laser

I present an analytical solution for the phase space evolution of electrons in a self-amplified spontaneous emission (SASE) free-electron laser (FEL) operating in the linear regime before saturation in the resonant case by solving the one dimensional FEL equation together with the solution of the cubic equation, which represents the evolution of the SASE FEL field. The electrons are shown to be bunched around $\ensuremath{\pi}/6$ ahead of a resonant electron every resonant FEL wavelength in the high gain regime. The phase relation is similar to that in a low gain FEL where an electron beam above resonance is injected, explaining the positive FEL gain. The analytical solutions agree well with numerical simulations and are applied to obtain the coherent optical transition radiation (OTR) intensity produced from electron microbunching at FEL wavelength. The coherent OTR intensity is shown to be proportional to FEL intensity.

On a fractional generalization of the free electron laser equation

The following fractional generalization of the free electron laser equation is investigated: D τ αh(τ)=λ∫ 0 τt δh(τ−t)Φ(b;δ+1; iνt) dt+μτ γΦ(β,γ+1; iντ), 0⩽τ⩽1, where β, γ, λ∈ C; ν, b, β∈ R, α>0, γ>−1 and δ>−1. A closed form solution is derived in terms of Kummer’s function Φ( α, β; z) by the application of Riemann–Liouville fractional integration operators. Tau method approximation is used in the evaluation of the results in a suitable form for numerical computation. The results derived are of general character and provide extension of the work reported by Boyadjiev et al. and Al-Shammery et al. [Math. Comput. Model. 25 (1997) 1; Integral Transform. Spec. Funct. 9 (2000) 81; J. Fract. Cal. Appl. Anal. 2 (1999) 501].

One-dimensional free-electron laser equations without slowly varying envelope approximation

It is shown how the usual first-order equations describing the one-dimensional free-electron laser dynamics can be obtained without relying on any a priori separation between fast and slow time and space variations in the radiation field (the well-known slowly varying envelope approximation). The exact first-order equations can be reduced to the approximate form and sufficient conditions are given for making this reduction. It is also shown that the start-up of the high-frequency signal is correctly described by the slowly varying approximation only when the electron bunch at t=0 has longitudinal density and/or parallel velocity fluctuations whose spectrum has significant values in the region comprising the wave number of the high-frequency wave.

Misalignment effects of segmented undulator in self-amplified spontaneous emission

Effects caused by misalignment of undulator segments on the gain of a free electron laser (FEL) based on the process of self-amplified spontaneous emission are theoretically studied. The well-known FEL equations are solved analytically and numerically with boundary conditions describing the misalignment effect in order to obtain the amplification gain as a function of the distance from the undulator entrance. The two worst cases of alignment are considered. One is that the phase slip over the drift section between the adjacent segments is $\ensuremath{\pi}$, and the other is that the radiation is cut off at the drift section. Calculations show that if the undulator segment is several times as long as the gain length, the radiation power is close to that of no segmentation in both cases.

Fractional free electron laser equations*

We are concerned with the fractional generalization of a Volterra-type integro-differential equation, known as a Free Electron Laser (FEL) equation. Different methods for solving a Cauchy-type initial value problem are discussed. The solutions of the one-dimensional fractional FEL (FFEL) equation are found in a closed form suitable for numerical treatment. Multidimensional FFEL equations are also considered and corresponding solutions are obtained in terms of Hermite and Laguerre polynomials. Numerical treatment of the results is also provided.