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The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.

Along with the development of FEL X-ray sources, the design and the construction of Compton Back-Scattering (CBS) devices for different applications are being pursued. These sources are designed to provide monochromatic, high peak brightness tunable light beams, with photon energies above 10 KeV. The source brightness is the most significant figure of merit, which specifies the suitability of X-ray beams for their application in a specific field of interest and also reflects the quality of the electron beam itself, which in turn depends on the charge, emittance, energy spread, etcetera. This article is intended as an introductory review to the field and we embed a few theoretical considerations and analytical formulae to develop simple tools for the design of CBS X-ray sources. The present attempt is analogous to previous efforts, put forward to model Free Electron Laser devices, using scaling relations accounting for the entanglement between the different physical quantities contributing to the output beam performances. We comment on the reliability of our analysis by a comparison with the wealth of results (either numerical and analytical) reported in the dedicated literature.

We discuss the general concepts underlying the design strategies of complex devices like Tokamaks and Free Electron Lasers (FEL). Regarding the FEL, starting from the desired output performances, the key parameters are embedded to get a set of semi-analytical/empirical equations yielding straightforward and reliable estimates of gain and power. In a similar way, the guiding elements of a fusion reactor, to reach the prescribed fusion gain Q and power, are defined in terms of scaling relations involving pivotal quantities like radius and magnetic field. General formulae characterizing a physical system may be the consequence of an unknown symmetry. The onset of specific instabilities represent the breaking of a symmetry characterizing given equilibrium conditions. In this article, we comment on the analogy between two different physical devices, and even though we do not specify any underlying symmetry, we aim to stimulate further research in this direction.

The umbral operational techniques offer sturdy mechanism in the studies of special functions and special polynomials. The techniques of the umbral calculus are employed to derive the properties of families of exponential like functions and their hyperbolic forms. The generalized forms of Mittag-Leffler functions are used to solve the technical problems concerning the transport of a charged beam in a solenoid magnet. The proposed method is flexible and has many advantages over the standard computational techniques.

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 in compact particle accelerators. The comparison, between the requirements imposed by these two concepts on the radiation sources on one hand and the output parameters of the currently available high-performance gyrotrons on the other hand, show that they match each other to a high degree. We consider this as a manifestation of the feasibility and potential of these concepts. It is believed that after the first successful proof-of-principle experiments they will find more wide usage in the advanced FEL and particle accelerators.

The FEL integral equation is reviewed here and is studied under different contexts, accounting for diverse physical regimes. We include higher order harmonics and saturation effects, and explain the origin of scaling relations, widely exploited to describe either FEL dynamics or nonnlinear harmonic generation.

The possibility of extending the tunability of Free-Electron Lasers towards short wavelengths has been explored through the design of devices conceived to enhance the mechanisms of nonlinear harmonic generation. In this respect, different schemes of operation have been suggested in the past, such as harmonic seeding, bi-harmonic undulators, and two-beam self-seeding devices. In this paper, we discuss how these methods can be merged into a tool, extending the performance of FEL devices.

Free Electron Lasers (FELs) are certainly among the most interesting devices, belonging to the realm of coherent radiation sources [...]

International audience; 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 performance. The achievement of chirped pulse amplification allowing lasers to be operated at the Terawatt range, opened the way for the Laser Plasma Acceleration (LPA) technique where high energy electron bunches with high current can be produced within a very short centimeter-scale distance. Such an advanced acceleration concept is of great interest to be qualified by an FEL application for compact X-ray light sources. We explore in this paper what the LPA specificities imply on the design of the undulator, part of the gain medium. First, the LPA concept and state-of-art are presented showing the different operation regimes and what electron beam parameters are likely to be achieved. The LPA scaling laws are discussed afterwards to better understand what laser or plasma parameters have to be adjusted in order to improve electron beam quality. The FEL is secondly discussed starting with the spontaneous emission, followed by the different FEL configurations, the electron beam transport to the undulator and finally the scaling laws and correction terms in the high gain case. Then, the different types of compact undulators that can be implemented for an LPA based FEL application are analyzed. Finally, examples of relevant experiments are reported by describing the transport beamline, presenting the spontaneous emission characteristics achieved so far and the future prospects.

The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed at enhancing the harmonic content of the emitted radiation. The analysis is eventually extended to the generalization of the FEL pendulum equation to treat Free Electron Laser operating with multi-harmonic undulators. The paper aims at picking out those elements supporting the usefulness of the Generalized Bessel Functions in the elaboration of the theory underlying the study of the spectral properties of the bremsstrahlung radiation emitted by relativistic charges, along with the relevant flexibility in accounting for a large variety of apparently uncorrelated phenomenolgies, like multi-photon processes, including non linear Compton scattering.

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism.

We discuss a hybrid Free Electron Laser (FEL) architecture operating with a RF undulator provided by a powerful Cyclotron Auto-Resonance Maser (CARM). We outline the design elements to operate a compact X-ray device. We review the essential aspects of wave undulator FEL theory and of CARM devices.

The use of non-standard calculus means have been proven to be extremely powerful for studying old and new properties of special functions and polynomials. These methods have helped to frame either elementary and special functions within the same logical context. Methods of Umbral and operational calculus have been embedded in a powerful and efficient analytical tool, which will be applied to the study of the properties of distributions such as Tsallis, Weibull and Student’s. We state that they can be viewed as standard Gaussian distributions and we take advantage of the relevant properties to infer those of the aforementioned distributions.

The use of non-local operators, defining Riemann–Liouville or Caputo derivatives, is a very useful tool to study problems involving non-conventional diffusion problems. The case of electric circuits, ruled by non-integer derivatives or capacitors with fractional dielectric permittivity, is a fairly natural frame of relevant applications. We use techniques, involving generalized exponential operators, to obtain suitable solutions for this type of problems and eventually discuss specific problems in applications.

European physical journal special topics 229(24), 3675 - 4284 (2020). doi:10.1140/epjst/e2020-000127-8, 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 research infrastructure based on novel plasma acceleration concepts and laser technology. It focuses on the development of electron accelerators and underlying technologies, their user communities, and the exploitation of existing accelerator infrastructures in Europe. EuPRAXIA has involved, amongst others, the international laser community and industry to build links and bridges with accelerator science — through realising synergies, identifying disruptive ideas, innovating, and fostering knowledge exchange. The Eu-PRAXIA project aims at the construction of an innovative electron accelerator using laser- and electron-beam-driven plasma wakefield acceleration that offers a significant reduction in size and possible savings in cost over current state-of-the-art radiofrequency-based accelerators. The foreseen electron energy range of one to five gigaelectronvolts (GeV) and its performance goals will enable versatile applications in various domains, e.g. as a compact free-electron laser (FEL), compact sources for medical imaging and positron generation, table-top test beams for particle detectors, as well as deeply penetrating X-ray and gamma-ray sources for material testing. EuPRAXIA is designed to be the required stepping stone to possible future plasma-based facilities, such as linear colliders at the high-energy physics (HEP) energy frontier. Consistent with a high-confidence approach, the project includes measures to retire risk by establishing scaled technology demonstrators. This report includes preliminary models for project implementation, cost and schedule that would allow operation of the full Eu-PRAXIA facility within 8—10 years., Published by Springer, Heidelberg

Self-amplified spontaneous emission (SASE) free electron laser (FEL) devices have disclosed an unexpected interplay between the laser intensity growth and regions of the electron bunch of the order of the coherence length. They are usually identified with the bunch slice and contribute to the laser dynamics with their own characteristics. The dynamical effects inducing geometrical and phase space misalignment of bunch slice in X-ray operating FELs can be traced back to a plethora of phenomena, both in the Linac accelerating section or inside the beam transport optic magnet. They are responsible for spoiling of the beam projected qualities and, if not corrected properly, induce an increase of the saturation length and a decreasing of the output power. We discuss the inclusion of these effects in models employing scaling formulae.

Effects producing gain dilution in Free Electron Laser devices are well documented. We develop here a unified point of view allowing the introduction of space charge effects, along with the gain deterioration due to inhomogeneous broadening contributions and discuss the relevant interplay. We outline future developments and comment on the possibility of including in the formalism effects of quantum mechanical nature., 10 pages, 7 figures

Over the past 90 years, particle accelerators have evolved into powerful and widely used tools for basic research, industry, medicine, and science. A new type of accelerator that uses plasma wakefields promises gradients as high as some tens of billions of electron volts per meter. This would allow much smaller accelerators that could be used for a wide range of fundamental and applied research applications. One of the target applications is a plasma-driven free-electron laser (FEL), aiming at producing tunable coherent light using electrons traveling in the periodic magnetic field of an undulator. In this work, the plasma-based electron beams with the most promising qualities, designed in the framework of EuPRAXIA, are analyzed in terms of the FEL performance.

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

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 formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well.

We study the small signal regime of Cyclotron Auto-Resonance Maser (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 two different numerical items: the home-made code GRAAL (Gyrotron Radiation Auto-Resonance Amplification Laser) and the, commercially available, software package CST. We prove the substantial consistence between the different methods and comment on the reasons leading to the discrepancies. The extension to the non-linear regime is finally touched on.

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

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 X rays in single spike mode with fluxes up to 10 12 electrons per shot have been found.

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 spectrum from microwaves to X-rays, and compares the analogies with conventional laser sources. We explore developing a point of view that allows a unified analytical treatment of these devices, by the introduction of appropriate global variables (e.g. gain, saturation intensity, inhomogeneous broadening parameters, longitudinal mode coupling strength), yielding a very effective way for the determination of the relevant design parameters. The paper looks also at more speculative aspects of FEL physics, which may address the relevance of quantum effects in the lasing process.

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 FEL, we analyze the quantum electrodynamics (QED) nature of the underlying laser mechanism. We look for the truly quantum signature in a process whose phenomenology is dominated by classical effects. As to the use of FEL as a tool for FP experiments we discuss the realization of a device dedicated to the study of non-linear effects in QED such as photon–photon scattering and shining-through-the-wall experiments planned to search for dark matter candidates like axions. © 2018 Elsevier B.V.

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 procedure to deal with applications in electron transport is also touched on.

We analyze different forms of light pulses with Orbital Angular Momentum based on two-index Hermite polynomials, their Poynting vector and angular momentum in paraxial propagation. We extend the analysis at first order in the paraxial approximation, calculating the transverse component of the Poynting vector and the longitudinal component of the angular momentum, which give the rotation of the energy flux around the vortices.

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., Comment: Corrected version

Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with 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 view for the repeated derivatives of sec ( . ) , tan ( . ) and for their hyperbolic counterparts.

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 efficient tool to explore the properties of these numbers.

International audience; The EuPRAXIA project aims at designing the world’s first accelerator based on plasma-wakefield advanced technique, which can deliver a 5 GeV electron beam with simultaneously high charge, low emittance and low energy spread to user’s communities. Such challenging objectives can only have a chance to be achieved when particular efforts are dedicated to identify the subsequent issues and to find the way to solve them. Many injection/acceleration schemes and techniques have been explored by means of thorough simulations in more than ten European institutes to sort out the most appropriate ones. The specific issues of high charge, high beam quality and beam extraction then transfer to the user’s applications, have been tackled with many innovative approaches*. This article highlights the different advanced methods that have been employed by the EuPRAXIA collaboration and the preliminary results obtained. The needs in terms of laser and plasma parameters for such an accelerator are also summarized.

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 nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions of fractional differential equations. We apply the method to a number of examples, in which fractional calculus and a certain umbral image calculus play a role of central importance., 16 pages, 2 figures

In self-amplified spontaneous emission free electron laser devices, the combined effect of slippage and energy spread impose significant limitations on the power performance. We provide a scaling formula capable of parameterizing the effect in terms of the electron beam parameters and discuss mechanisms leading to the output power optimization. © 2019 SPIE.

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 functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of SJ polynomials explicitly., 23+11 pages, 2 tables; MDPI LaTeX class, published version

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 of a 250 GHz, 0.5 MW CARM has been undertaken at ENEA. Mode converters from the TE10 in WR3 rectangular waveguide to the operational mode (TE53) are required to perform the RF cold test of Bragg reflectors and cavities. In this paper an improved and feasible structure for the TE11 to TE04 conversion, in oversized circular waveguides, using only two transitions with profile and radius optimized in order to obtain high efficiencies and sufficient bandwidth, has been designed and the simulation results are presented. The first transition is a TE11/TE01 serpentine mode converter in circular waveguide (efficiency of 95%) with average radius of 1.48 mm and an appropriate number of geometrical periods. The second one, a TE01/TE04 rippled wall mode converter with efficiency of 97%, consists of a single conversion section with an average radius of 3 mm and sixteen sinusoidal periods instead of three conversion sections (TE01/TE02, TE02/TE03, TE03/TE04) in cascade, thus reducing the complexity and the length of the full transition chain.

We study the time-dependent solutions of Schrodinger 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 spatial variables. Here, we reconsider the integration method by the use of the formalism of two-variable Hermite polynomials providing a very simple derivation of the relevant physical quantities. This method is eventually exploited to study different problems including anharmonic oscillators and pendulum-like potentials. Regarding the case of periodic potentials, we touch on the application of the method to the quantum free-electron laser dynamics. Finally, we comment on future developments of this line of research regarding the relevant comparison with other exact and approximate integration schemes.

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 approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper., Comment: 19 pages, 5 figures

25th International Conference on the Application of Accelerators in Research and Industry, CAARI 2018, Texas, USA, 12 Aug 2018 - 17 Aug 2018; AIP conference proceedings 2160(1), 040012-1 - 040012-9 (2019). doi:https://doi.org/10.1063/1.5127692, 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 with eXcellence In Applications”) aims to overcome the first three of these hurdles by developing a conceptual design for a first international user facility based on plasma acceleration. In this paper we report on the main features, simulation studies and potential applications of this future research infrastructure., Published by AIP Publishing, Melville, NY

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 capable of parameterizing the effect in terms of the device parameter and discuss possible mechanisms allowing the power output optimization. © 2018 Elsevier GmbH

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 algebraic properties. Within such a respect we derive generalized forms of Euler exponential formula and explore the usefulness and relevance of operator ordering procedure of the Wei-Norman type. We also discuss the properties of dual numbers in terms of Pauli matrices. Finally we explore generalized forms of Dirac-like factorization, emerging from the properties of these numbers.

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 α (0 < α < 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 time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering. © 2018 Diogenes Co., Sofia.

We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired to the Bessel Calculus of Bochner, Cholewinsky and Haimo. We propose further extensions of the method and of the relevant concepts as well and obtain new families of integral transform allowing the framing of the previous concepts within the context of generalized Borel tranforms., Comment: 15 pages and 3 figs

We study functions related to the experimentally observed Havriliak- Negami dielectric relaxation pattern proportional in the frequency domain to [1 + (iωτ0)α]-β with τ0 > 0 being some characteristic time. For α = l/k < 1 (l and k being positive and relatively prime integers) and β > 0 we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their domain dual in the sense of the inverse Laplace transform. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization β = (2 - q)/(q - 1) and τ0 = (q - 1)1/α (1 < q < 2) we show that for 0 < α < 1 the response functions fα,β(t/τ0) go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities gα,β (u), we introduce two-variable densities and show that they satisfy the integral form of the evolution equation. © 2018 IOP Publishing Ltd.