A note on the exact solution for the magnetic field in a solenoid
We comment on a recent paper regarding the derivation of the magnetic field components of a solenoid in analytical form by proposing a different and simpler method
physics.acc-ph↗
arXiv subjects
E. Sabia
Publications and source records attributed to E. Sabia.
Umbral-Algebraic Methods and Asymptotic Properties of Special Polynomials
A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial families, with the use of elementary algebraic tools.
math.CA↗
Products of Bessel functions and associated polynomials
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.
math-ph↗
Operational versus umbral methods and the Borel transform
Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.
math.CA↗
EuPRAXIA@SPARC_LAB Design study towards a compact FEL facility at LNF
On the wake of the results obtained so far at the SPARC\_LAB test-facility at the Laboratori Nazionali di Frascati (Italy), we are currently investigating the possibility to design and build a new multi-disciplinary user-facility, equipped with a soft X-ray Free Electron Laser (FEL) driven by a $\sim$1 GeV high brightness linac based on plasma accelerator modules. This design study is performed in synergy with the EuPRAXIA design study. In this paper we report about the recent progresses in the on going design study of the new facility.
physics.acc-ph↗
Theory of relativistic heat polynomials and one-sided Lévy distributions
The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper we report on new and unsuspected links between fractional diffusion, quantum relativistic equations and particular families of polynomials, linked to the Carlitz family, and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance.
math-ph↗
Deep Saturated Free Electron Laser Oscillators and Frozen Spikes
We analyze the behavior of Free Electron Laser (FEL) oscillators operating in the deep saturated regime and point out the formation of sub-peaks of the optical pulse. They are very stable configurations, having a width corresponding to a coherence length. We speculate on the physical mechanisms underlying their growth and attempt an identification with FEL mode locked structures associated with Super Modes. Their impact on the intra-cavity nonlinear harmonic generation is also discussed along with the possibility of exploiting them as cavity out-coupler.
physics.acc-ph↗
Pathway to a Compact SASE FEL Device
Newly developed high peak power lasers have opened the possibilities of driving coherent light sources operating with laser plasma accelerated beams and wave undulators. We speculate on the combination of these two concepts and show that the merging of the underlying technologies could lead to new and interesting possibilities to achieve truly compact, coherent radiator devices.
physics.acc-ph↗
Laser Heater and seeded Free Electron Laser
In this paper we consider the effect of laser heater on a seeded Free Electron Laser. We develop a model embedding the effect of the energy modulation induced by the heater with those due to the seeding. The present analysis is compatible with the experimental results obtained at FERMI displaying secondary maxima with increasing heater intensity. The treatment developed in the paper confirms and extends previous analyses and put in evidence further effects which can be tested in future experiments.
physics.acc-ph↗
Relativistic Wave Equations: An Operational Approach
The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schrödinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.
math-ph↗
Bunching coefficients in Echo-Enabled Harmonic Generation
Coulombian diffusion determines a dilution of bunching coefficients in Free Electron Laser seeded devices. From the mathematical point of view the effect can be modeled through a heat type equation, which can be merged with the ordinary Liouville equation, ruling the evolution of the longitudinal phase space beam distribution. We will show that the use of analytical tools like the Generalized Bessel Functions and algebraic techniques for the solution of evolution problems may provide a useful method of analysis and shine further light on the physical aspects of the underlying mechanisms.
physics.acc-ph↗
Relativistic harmonic oscillator
We consider the relativistic generalization of the harmonic oscillator problem by addressing different questions regarding its classical aspects. We treat the problem using the formalism of Hamiltonian mechanics. A Lie algebraic technique is used to solve the associated Liouville equations, yielding the phase space evolution of an ensemble of relativistic particles, subject to a "harmonic" potential. The non-harmonic distortion of the spatial and momentum distributions due to the intrinsic non-linear nature of the relativistic contributions are discussed. We analyze the relativistic dynamics induced by two types of Hamiltonian, which can ascribed to those of harmonic oscillators type. Finally, we briefly discuss the quantum aspects of the problem by considering possible strategies for the solution of the associated Salpeter equation.
math-ph↗
Chebyshev polynomials and generalized complex numbers
The generalized complex numbers can be realized in terms of $2\times2$ or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers. We also briefly discuss the two-variable Chebyshev polynomials and their link with the third-order Hermite polynomials.
math.CA↗
Complex-type numbers and generalizations of the Euler identity
We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural way in algebraic and geometric terms.
math.CA↗
Slices and Ellipse Geometry
We discuss the new problems emerging in charged beam transport for SASE FEL dynamics. The optimization of the magnetic transport system for future devices requires new concepts associated with the slice emittance and the slice phase space distribution. We study the problem of electron beam slice matching and guiding in transport devices for SASE FEL emission discussing matching criteria and how the associated design of the electron transport line may affect the FEL output performances. We analyze different matching strategies by studying the relevant effect on the FEL output characteristics.
physics.acc-ph↗
The FEL SASE operation, bunch compression and the beam heater
We discuss the conditions required for an optimal SASE FEL operation when bunch compression techniques are exploited to enhance the bunch peak current. We discuss the case of velocity bunching and magnetic bunch compression. With the reference to the latter technique we provide a quantitative estimate of the amount of laser heater power necessary to suppress the micro-bunching instability without creating any problem to the SASE dynamics
physics.acc-ph↗
Slice emittance, projected emittance and properties of the FEL SASE radiation
The existence of a characteristic coherence length in FEL SASE Physics determines the independent lasing of different portions, namely the slices, of the electron bunch. Each slice may be characterized by different phase space properties (not necessarily equal emittances and Twiss coefficients). This fact opens new questions on the concept of beam matching and how the various portions of the beam contribute to the performances of the output radiation, including those associated with the transverse coherence.
physics.acc-ph↗
Generalized Transforms and Special Functions
We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A general method, based on the use of the Fourier transform technique, is proposed for the study of the properties of functions of operators.
math-ph↗