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.


I. INTRODUCTION
In this paper we will pursue some technical details concerning the computation and the physical understanding of the recent analysis by Stupakov [1] on the effect of Coulomb diffusion on bunching coefficients in Echo Enabled Harmonic Generation (EEHG) Free Electron Laser (FEL) seeded devices [2]. The considerations developed in this paper should be understood as a complement to refs. [1,2], with the aim of providing a more general computational framework, benefitting from the formalism of beam transport employing algebraic means [3].
In this treatment the transport through magnetic elements is treated in terms of exponential operator acting on an initial phase space distribution and usually do not contain "transport" elements provided by a heat type diffusion. Here we will show that diffusion mechanisms can be included in such a framework in a fairly straightforward way [4], preserving the spirit of exponential operator concatenation.
The effect of the Coulomb diffusion on the e-beam longitudinal phase space distribution is ruled, according to [1], by the equation Where s is the coordinate of propagation, ζ is the longitudinal coordinate in the beam, p is associated to the beam energy, D is the Coulomb diffusion coefficient and f 0 (p, ζ) is the "initial" distribution. We will discuss the physical meaning of the previous quantities and we will provide their explicit expression in the following, for the moment we note that, having assumed that D is independent of p , we can solve (1.1) via the following Gauss-Weierstrass The bunching coefficients can be defined as those of the Fourier components of the expansion [6] f (p, ζ, s) = +∞ n=−∞ b n (p, s) e inζ (1.3) and are, therefore, specified by The dilution of the bunching coefficients induced by the Coulomb diffusion can therefore be mathematically expressed as Following ref. [1], we assume the initial distribution (1.6) whose physical meaning will be discussed in the forthcoming section. The solution of eq.
(1.1) can be therefore easily computed, even though cannot be obtained in analytical form.
In Fig. 1 we have reported the effect of the dispersion on the initial distribution (Fig. 1a), after a drift s = 0.3 with D = 0.7 (Fig. 1b). It is evident that the Coulomb term smears out the oscillations, associated with the bunching coefficients, which tend to disappear or to be strongly suppressed, as we will further discuss in the following. If we neglect in eq. (1.6) the contributions (A 1 sin (ζ − B 1 p)) 2 we can expand f 0 (p, ζ) in series of Bessel functions, namely Where J n (·) are cylindrical Bessel functions of the first kind. The associated bunching coefficients are, neglecting contributions in The G-W transform can be accordingly written as (See Appendix) and bunching coefficients now read The effect of the Coulombian diffusion is therefore twofold, it induces a) A dispersion due to the term 2Ds b) A suppression of the higher orders harmonics occurring through e − (mB 1 ) 2 Ds
In these introductory remarks we have provided a few remarks on the type of formalism we are going to use to treat the bunching mechanism in EEHG FEL seeded devices, the forthcoming sections will cover more physical details.

DYNAMICS
We have quoted the initial distribution given in eq. (1.6) without any comment about its physical meaning. Its origin can be traced back to the following Liouville equation which rules the evolution of an ensemble of non-interacting particles, driven by the Hamil- The solution of eq. (2.1) can be cast in the form We have denoted byÛ (s) andL the evolution and Liouville operators, respectively, of our dynamical problem [3,4]. We note that the Liouville operator breaks into two noncommuting parts, therefore any treatment of the associated evolution operator demands for an approximate disentanglement of the exponential, which, for simplicity will be assumed to be provided by [3,4] (2.5) Therefore by setting and keep as initial distribution a Gaussian in p , we recover eq. (1.6).
The physical scheme, we are dealing with, is reported in ref. [2]), to which the reader is addressed for further details. The e-beam undergoes two successive modulations induced by two different lasers in two different sections. The beam initial distribution (1.6) is that occurring after the first chicane at the entrance of the second modulator.
The physical content of the previous variables is the following where σ E is the e-beam energy spread, ∆E 1 is the induced energy modulation, k L is the laser wave vector.
The dynamics of the bunching coefficients can be obtained from eqs. (2.1) and (1. Eq. (2.8) yields an idea of the coupling between the various bunching coefficients and has already been derived in a closely similar fom in ref. [6].
The inclusion of the Coulomb diffusion in this scheme can be modeled by transforming the Liouville into a Vlasov equation, namely The diffusion coefficient, expressed in practical units, reads [1] with ε x and σ x being the emittance and beam section respectively The solution of eq. (2.9) can be obtained as f (p, ζ, s) ∼ = e Ds∂ 2 p (e sL f 0 (p, ζ)) (2.11) According to this approximation the effect of the diffusion is calculated separately from that induced by the Liouvillian contribution. Higher order disentanglements [3,4] can be used to get more accurate results, but the present approximation is adequate for our purposes.
In Fig. 2 we report the contour plots of the Liouville distribution with and without the effect of the Coulombian diffusion for the same s value. It is evident that the presence of a non-zero D value provides a significant reduction of the distribution harmonic content.
A global view is provided by Fig. 3 where we have reported the 3D plot of the Liouville distribution under the action of diffusion for different s-values. The nice feature of the procedure we are adopting is its modularity, which will be further appreciated in the forthcoming section.

III. FINAL COMMENTS
In the introductory section we have expanded the initial distribution, by neglecting the contributions in A 2 . Such an approximation implies that the induced energy modulation is not large compared to the natural e-beam energy spread. However if we relax these assumptions and employ the formalism of generalized Bessel functions [5], we can obtain a more general view on the problem under study. It has indeed been shown that two variable Bessel functions can be expressed through the generating function H n+2r (x, y) r!(n + r)! , x n−2r y r r!(n − 2r)! where H n (x, y) are two variable Hermite-Kampï¿oe dï¿oe Fï¿oeriï¿oet polynomials [5].
They are actually understood as belonging to the family of Hermite based functions and are widely exploited to deal with radiation scattering problems [7], going beyond the dipole approximation.
It is also worth noting the following operational property [5] In the second modulator the initial distribution will be provided by the properly modified eq. (1.9) so that which holds if the laser, in the second modulator, is the same of the first. We have removed the s variable because absorbed into the A and B terms. If we use the Bessel function we can therefore cast the distribution at the end of the second modulator as By rearranging the indices we end up with According to such a picture the n-th bunching coefficient is characterized by a discrete convolution of the bunching terms of the second modulator on the first.
The effect of the Coulombian diffusion on the distribution (3.6) can be easily computed, either numerically or analytically. By keeping the lowest order contribution l = 1 and by approximating the Bessel function as we obtain From the above equation we easily deduce that the Coulombian contribution yields a suppression factor proportional to e − (B 1 +2B 2 ) 2 Ds

1+4Ds
. More in general as also pointed in ref. [1] the larger is the order of bunching the more significant is the effect of the reduction.
The bunching coefficients vs. s in the presence of diffusion is provided by Fig. 4. The formalism we have presented in this paper provides a fairly detailed analysis of how the Coulombian diffusion affect the bunching coefficients in Echo enabled FEL devices. Our treatment is complementary to the seminal contributions of refs. [1,2] and confirm, within a more general framework, their conclusions. We believe that the modularity of the methods we have described in this paper offer further opportunities as e. g. that of including in the treatment more complicated arrangements of transport lines and of external fields. The possibility of embedding these procedures with diffusion and damping in storage rings, will be discussed elsewhere.

Appendix
We will discuss here some technical details concerning the solution of the heat equation F (x, 0) = x n e −x 2 which can formally be written as The use of operational methods [5] yields F (x, t) = (e t∂ 2 x x n e −t∂ 2 x )e t∂ 2 x e −x 2 = = (x + 2t∂ x ) n e t∂ 2 x e −x 2 .
Then by applying the Glaisher and Burchnall [5] rules, respectively reported below It is accordingly evident that for any initial condition such that