Giuseppe Dattoli

ENEA | ENEA · Frascati Research Centre

Gyrotrons are used as high-power sources of coherent radiation operating in pulsed and CW regimes in many scientific and technological fields. In this paper, we discuss two of their numerous applications. The first one is in gyrotron-powered electromagnetic wigglers and undulators. The second one is for driving high-gradient accelerating structures...

The report presents, as the main result of the CompactLight project, the conceptual design of the CompactLight hard X-ray FEL. It is divided in the following chapters: 1. Executive Summary 2. Introduction 3. Science Goals and Photon Output Requirements 4. Systems Design and Performance 5. Accelerator 6. Light Production 7. Civil Engineering...

The fourth generation of synchrotron radiation sources, commonly referred to as the Free Electron Laser (FEL), provides an intense source of brilliant X-ray beams enabling the investigation of matter at the atomic scale with unprecedented time resolution. These sources require the use of conventional linear accelerators providing high electron beam...

We derive integrals of combination of Gauss and Bessel functions, by the use of umbral techniques. We show that the method allows the possibility of pursuing new and apparently fruitful avenues in the theory of special functions, displaying interesting links with the theory and the formalism of integral transforms.

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural a...

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

This report presents the conceptual design of a new European research infrastructure EuPRAXIA. The concept has been established over the last four years in a unique collaboration of 41 laboratories within a Horizon 2020 design study funded by the European Union. EuPRAXIA is the first European project that develops a dedicated particle accelerator r...

This report presents the conceptual design of a new European research infrastructure EuPRAXIA. The concept has been established over the last four years in a unique collaboration of 41 laboratories within a Horizon 2020 design study funded by the European Union. EuPRAXIA is the first European project that develops a dedicated particle accelerator r...

Figure 20.1 was not correct in the published article. The original article has been corrected. The published apologizes for the inconvenience.

In a previous note we made an analysis of the spreading of the COVID disease in Italy. We used a model based on the logistic and Hubbert functions, the analysis we exploited has shown limited usefulness in terms of predictions and failed in fixing fundamental indications like the point of inflection of the disease growth. In this note we elaborate...

Different forms of trigonometry have been proposed in the past to account for geometrical and applicative issues. Along with circular trigonometry, its hyperbolic counterpart has played a pivotal role to provide the geometrical framework of special relativity. The parabolic trigonometry is in between the previous two, and we discuss the relevant pr...

The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral representations linking Hermite and Legendre like polynomials and of operational technique allow the derivation of new pro...

Differintegral methods, namely those techniques using differential and integral operators on the same footing, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated...

We present two types of polynomials related to the Mittag-Leffler function namely the fractional Hermite polynomial and the Mittag-Leffler polynomial. The first modifies the Hermite polynomial and the second one is a refashioned Laguerre polynomial. The fractional Hermite and the Mittag-Leffler polynomials are used to solve {the Cauchy problems for...

The Horizon 2020 project EuPRAXIA (European Plasma Research Accelerator with eXcellence In Applications) is producing a conceptual design report for a highly compact and cost-effective European facility with multi-GeV electron beams accelerated using plasmas. EuPRAXIA will be set up as a distributed Open Innovation platform with two construction si...

The EuPRAXIA project aims at designing the world's first accelerator based on advanced plasma-wakefield techniques to deliver 5 GeV electron beams that simultaneously have high charge, low emittance and low energy spread, which are required for applications by future user communities. Meeting this challenging objective will only be possible through...

Differintegral methods, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and o...

Plasma accelerators present one of the most suitable candidates for the development of more compact particle acceleration technologies, yet they still lag behind radiofrequency (RF)-based devices when it comes to beam quality, control, stability and power efficiency. The Horizon 2020-funded project EuPRAXIA (“European Plasma Research Accelerator wi...

Electron Cyclotron Resonance Heating (ECRH) systems in future fusion devices, like the DEMO-nstration reactor, foresee an operational frequency in the range 230–280 GHz to match the plasma characteristics. Cyclotron Auto Resonance Masers (CARMs) could represent an alternative to gyrotrons as effective source of mm and sub-mm waves and the project o...

Differintegral methods, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and o...

CompactLight (XLS) is an International Collaboration of 24 partners and 5 third parties, funded by the European Union through the Horizon 2020 Research and Innovation Programme. The main goal of the project, which started in January 2018 with a duration of 36 months, is the design of an hard X-ray FEL facility beyond today's state of the art, using...

Elementary problems like the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated by the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of vie...

The H2020 CompactLight Project aims at designing the next generation of compact X-rays Free-Electron Lasers, relying on very high gradient accelerating structures (X-band, 12 GHz), the most advanced concepts for high brightness electron photo injectors, and innovative compact short-period undulators. Compared to existing facilities, the proposed fa...

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficien...

The q-calculus is reformulated in terms of the umbral calculus and of the associated operational formalism. We show that new and interesting elements emerge from such a restyling. The proposed technique is applied to a different formulations of q special functions, to the derivation of integrals involving ordinary and q-functions and to the study o...

We embed integral transform methods and operational techniques to derive and generalize some results concerning the Legendre polynomials P n (x) regarding the possibility of expressing P n (λx), with λ positive constant, as a sum of Legendre polynomials. The technique we propose is shown to be profitably extended to other families of special polyno...

The book is ntended as a series of lectures on the physics and technology of free electron lasers

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper a natural ap...

Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

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 combined use of umbral and operational methods. The technique we propose reduces the derivation of old and new asymptotic identities to the straightforward application of elementary calculus rules.

Dual Numbers and their higher order version turned out to be an important tool for 1 numerical computations, and in particular for finite difference calculus. We will use the relevant 2 algebraic rules and matrix realization to present a novel point of view, embedding dual numbers 3 within a formalism reminiscent of operational umbral calculus. 4

The use of operational methods of different nature is shown to be a fairly powerful tool to study different problems regarding the theory of Legendre and Legendre-like polynomials. We show how the use of the well known integral representations linking Hermite and Legendre like polynomials and of operational technique allow the derivation of new pro...

We derive infinite integrals of combination of Gauss and Bessel functions, by the use of umbral techniques. We show that the method allows the possibility of pursuing new and apparently fruitful avenues in the theory of special functions, displaying interesting links with the theory and the formalism of fractional calculus.

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational na...

We study the time-dependent solutions of Schrödinger equations ruled by different non-singular potentials. We employ a recently proposed integration procedure, assuming a time-dependent Gaussian shape for the wave function. The method is independent of the specific form of the potential and allows a straightforward separation of the time and spatia...

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational na...

The Hermite and Laguerre functions are the extension of the corresponding polynomials to negative and/ or real indices. We show that the use of methods of operational nature allows the relevant definition within a straightforward and unified fashion. It is illustrated how the techniques, we are going to suggest, provide an easy derivation of the re...

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric...

A family of two variable polynomials naturally emerges from the expansion in multipoles of a magnetic field. The order of the expansion fixes the polynomial degree and the multipole content: dipole, quadrupole, sextupole,... The associated polynomials share analogies with Hermite-type families. We take advantage from this analogy to present a fairl...

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 combined use of umbral and operational methods. The technique we propose reduces the derivation of old and new asymptotic identities to the straightforward application of elementary calculus rules.

We study the small signal regime of CARM, using different computational tools.The analysis is motivated by the necessity of an accurate benchmarking of the design parameter for the CARM device under construction at ENEA Frascati center. The main goal is that of comparing the results from three different procedures, the analytical formulation and tw...

The combined effect of slippage and energy spread in self-amplified-spontaneous-emission Free Electron Laser devices, operating in single spike regime, may impose limitations on the relevant performance. If the slippage is larger than the electron bunch length, the saturated power is significantly reduced. We provide a simple scaling formula capabl...

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric...

We use the results of a recent reformulation of the theory of arbitrary order differential equations in terms of non-Hermitian operators to show that the invariant binorm is associated to a generalized Courant-Snyder invariant. Furthermore , we indicate the existence of higher-order invariants associated to the Casimir operators of the group, utili...

The efficiency of Free Electron Laser (FEL) Oscillator devices is a fairly complicated function of the various parameters which characterize the device itself. We explore the relevant dependences by the use of the scaling formulas describing the FEL Oscillator dynamics and providing the relevant design elements. We obtain a quantitative dependence...

Different forms of trigonometry have been proposed in the past to account for geometrical and applicative issues. Along with circular trigonometry, its hyper-bolic counterpart has played a pivotal role to provide the geometrical framework of special relativity. The parabolic trigonometry is in between the previous two, and we discuss the relevant p...

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.

The use of Minkowsky diagram for an introduction to special relativity is discussed, along with the possibility of exploiting the hyperbolic trigonometry to explore some puzzling features of special relativity.

We present an analysis of the FEL performances of several working points belonging to three different classes of electron beams driven by the same infrastructure: two high flux working points are relevant to the operation with a linac up to 1 GeV, while both LWFA and PWFA beams have been considered at this same energy for low charge operation. Soft...

The theory of harmonic based function is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.

In this paper we review the notion of hybrid complex numbers, recently introduced to provide a comprehensive conceptual and formal framework to deal with circular, hyperbolic and dual complex. We exploit the established isomorphism between complex numbers as abstract entities and as two dimensional matrices in order to derive the associated algebra...

LHC ha confermato molte previsioni teoriche, che datavano alla seconda metà del 900. A parte la più eclatante, ovvero la prova dell'esistenza del bosone BEH (Brout-Englert-Higgs), altre, se pure con minore enfasi mediatica perché note solo agli addetti ai lavori, stanno completando un quadro teorico che si era consolidato tra gli anni 70 e 80. In p...

We introduce higher order harmonic numbers and derive the relevant properties and generating functions by the use of an umbral type technique. We discuss the link with recent works on the subject and show that the combination of umbral and other techniques (like Laplace and other types of integral transforms) yields a very efficient tool to explore...

In this paper we review the notion of hybrid complex numbers, recently introduced to provide a comprehensive conceptual and formal framework to deal with circular, hyperbolic and dual complex. We exploit the established isomorphism between complex numbers as abstract entities and as two dimensional matrices in order to derive the associated algebra...

We propose a simple way of combining Lucas and Fibonacci numbers, to get families of discrete trigonometric functions, including circular and hyperbolic types. Their use in application is discussed along with the link to previous investigations.We explore the relevant elementary properties including addition and multiplication theorems, viewed as g...

We study functions related to the experimentally observed Havriliak–Negami dielectric relaxation pattern proportional in the frequency domain to {$[1+({\rm i}\omega\tau_{0}){\hspace{0pt}}^{\alpha}]^{-\beta}$} with {$\tau_{0} > 0$} being some characteristic time. For {$\alpha = l/k< 1$} ( l and k being positive and relatively prime integers) and {$\...

The main object of this paper is to show that combined use of the Lagrange expansion and certain operational techniques allows to derive mixed generating functions of various families of generalized polynomials in a straightforward manner. Relevant connections with many other recent works on this subject are also discussed.

The term Free Electron Laser (FEL) will be used, in this paper, to indicate a wide collection of devices aimed at providing coherent electromagnetic radiation from a beam of “free” electrons, unbound at the atomic or molecular states. This article reviews the similarities that link different sources of coherent radiation across the electromagnetic...

We present the conceptual design for a Cyclotron Auto Resonance Maser source, operating at 250 GHz and conceived for Plasma Fusion research activities. The study discussed here is aimed at planning the construction of such a device at the ENEA Frascati Center, within the framework of the researches pertaining to the Fusion department. This foreseen...

This review paper is devoted to the understanding of free-electron lasers (FEL) as devices for fundamental physics (FP) studies. After clarifying what FP stands for, we select some aspects of the FEL physics which can be viewed as fundamental. Furthermore, we discuss the perspective uses of the FEL in FP experiments. Regarding the FP aspects of the...

We present an analysis of the FEL performances of several working points belonging to three different classes of electron beams driven by the same infrastructure: two high flux working points are relevant to the operation with a linac up to 1 GeV, while both LWFA and PWFA beams have been considered at this same energy for low charge operation. Soft...

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 acce...

We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $\alpha$ ($0 < \alpha < 1$) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the tim...

We have considered a novel scheme of wave undulator FEL. The system employs a recirculated radiation pulse serving as undulator provided by a high power laser. Non conventional electron acceleration schemes promise nowadays high gradient acceleration yielding the GeV on the scale of few centimeters, however these solutions might solve the problem o...

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 sol...

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 brid...

The Generalized Trigonometric Functions (GTF) have been introduced using an appropriate redefinition of Euler type identities involving non-standard forms of imaginary numbers, realized by different types of matrices. In this paper we use the GTF to get parameterization of practical interest for non-singular matrices. The possibility of using this...

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which all...