2. PHENOMENOLOGICAL MODEL OF SASE FEL

The analysis of spontaneous UR with account for major sources of spectral line broadening is best done with the help of the generalized special functions of Bessel and Airy type, which describe respectively the UR spectrum and the shape of the UR lines, as done, for example, by Dattoli et al. (Reference Dattoli, Mikhailin and Zhukovsky2008, Reference Dattoli, Mikhailin and Zhukovsky2009), Mikhailin et al. (Reference Mikhailin, Zhukovsky and Kudiukova2014), Zhukovsky (Reference Zhukovsky2014a, Reference Zhukovskyb, Reference Zhukovskyc), and give exact account for the finite size of the beam, its divergence, energy spread, etc. (see, e.g., Quattromini et al. (Reference Quattromini, Artioli, Di Palma, Petralia and Giannessi2012); Zhukovsky (Reference Zhukovsky2015c, Reference Zhukovsky2016b)). As regards SASE FEL dynamics, basic equations for its analytical model were developed by Dattoli et al. (Reference Dattoli, Giannessi, Ottaviani and Ronsivalle2004, Reference Dattoli, Ottaviani and Renieri2005a, Reference Dattoli, Ottaviani and Pagnuttib), and applied also in the common work with Dattoli et al. (Reference Dattoli, Mikhailin, Ottaviani and Zhukovsky2006a). However, the above-cited works contain some inconsistencies and misprints, which frequently yield unphysical results, if the formulae are applied literally. In particular, it regards nonlinear regime for high harmonics, especially in segmented devices, and account for the losses. We remind that the wavelength of the radiated nth UR harmonic of a common planar undulator reads as follows:

(1)$${\rm \lambda} _n = \displaystyle{{{\rm \lambda} _0} \over n},\quad {\rm \lambda} _0 = \displaystyle{{{\rm \lambda} _{\rm u}} \over {2{\rm \gamma} ^2}} \left( {1 + \displaystyle{{k^2} \over 2}} \right),$$

where γ is the relativistic parameter, λ_u is the undulator period, and k is the undulator parameter

(2)$$k = {{eH_0 {\rm \lambda} _{\rm u}} / {2{\rm \pi} mc^2}}, \quad k^2 = 2{\rm \gamma} ^2 {\rm \beta} _ \bot ^2,$$

β_⊥ is the electron velocity component, normal to the undulator axis, H ₀ is the strength of the undulator magnetic field, e and m are the electron charge and mass, and c is the speed of light. Consider the planar two-frequency undulator of N periods with the following magnetic field:

(3)$$\eqalign{& \vec H = H_0 ({\rm \rho}, {\rm \kappa} + \sin (k_{\rm \lambda} z) + d\sin (hk_{\rm \lambda} z),0), \cr & \quad\quad h \in {\rm Z,}\quad {\rm \rho}, {\rm \kappa} = {\rm const}{\rm.}} $$

The non-periodic magnetic components $H_x^{{\rm const}} = {\rm \rho} H_0 $ and $H_y^{{\rm const}} = {\rm \kappa} H_0 $ yield the effective bending angle ${\rm \theta} _{\rm H} = (2/\sqrt{3})$ $(k/{\rm \gamma}) {\rm \pi} {\kern 1pt} N{\kern 1pt} {\rm \kappa} _1 $, which depends on the total strength of the constant magnetic component via the parameter ${\rm \kappa} _1 = \sqrt {{\rm \kappa} ^2 + {\rm \rho} ^2} $. Its influence is similar to that of the off-axis deviation in the angle θ_H. The UR intensity with account for the divergence reads (see Zhukovsky, Reference Zhukovsky2016b) as follows:

(4)$$\eqalign{& \left. {\displaystyle{{d^2 I} \over {d{\rm \omega} d\Omega}}} \right \vert _{(H_{\rm d} /H_0 )^2 \ll 1/(4{\rm \pi} N)^2} \cong \displaystyle{{e^2 N^2 {\rm \gamma} ^2} \over c}\displaystyle{{k^2} \over {(1 + {{k_{{\rm eff}}^2} / 2})^2}} \cr & \quad \times \sum\limits_{n = 1}^\infty {n^2 \left[ {S\left( {{\rm \alpha}, {\rm \beta}, {\rm \eta}} \right)\left( {T_{n - 1} + T_{n + 1} + \displaystyle{d \over h}\left( {T_{n + {\rm h}} + T_{n - {\rm h}}} \right)} \right)} \right]^{\,2}},}$$

where $k_{{\rm eff}}^2 = k^2 + k_2^2 $, k ₂ = k|d/h|, and it depends on the generalized Airy-type function $S({\rm \alpha}, {\rm \beta}, {\rm \eta} ) \equiv \int_0^1 {d{\rm \tau} e^{i({\rm \alpha} {\rm \tau} + {\rm \eta} {\rm \tau} ^2 + {\rm \beta} {\rm \tau} ^3 )}} $, as demonstrated, for example, in the recent work by Zhukovsky (Reference Zhukovsky2015c), and on the generalized Bessel functions

(5)$$\eqalign{& T_n = \int_0^{2{\rm \pi}} {\displaystyle{{d{\rm \varphi}} \over {2{\rm \pi}}}} \cos n \cr & \quad \quad \left[ \matrix{{\rm \varphi} + \displaystyle{{k^2 \sin (2{\rm \varphi} )} \over {4(1 + k_{{\rm eff}}^2 /2)}} - \displaystyle{{dk^2 \sin ((h - 1){\rm \varphi} )} \over {h(h - 1)(1 + k_{{\rm eff}}^2 /2)}} - \cr \displaystyle{{dk^2 \sin ((h + 1){\rm \varphi} )} \over {h(h + 1)(1 + k_{{\rm eff}}^2 /2)}} - \displaystyle{{d^2 k^2 \sin (2h{\rm \varphi} )} \over {4h^3 (1 + k_{{\rm eff}}^2 /2)}}} \right].}$$

The non-periodic magnetic components $H_x^{{\rm const}} = {\rm \rho} H_0 $, $H_y^{{\rm const}} = {\rm \kappa} H_0 $ are accounted for by the parameter β = ((2πnN  +  ν_n)(γθ_H)²/1  + (k²/2)(1 + (d/h)²) + (γθ_H)²), ν_n = 2πNn((ω/ω_n) − 1) is the detuning parameter, the off-axis angles Θ_x,y interplay with κ₁ in the parameter η = ((2πN)² nkγ/1 + k ²/2 + (γψ)²)(κΘ_x − ρΘ_y), where ${\rm \theta} _{\rm H} = (2/\sqrt{3})(k/{\rm \gamma}){\rm \pi} {\kern 1pt} N{\rm \kappa} _1 $, and the sum of their squares constitutes the angle ${\rm \psi} = \sqrt {\Theta _x^2 + \Theta _y^2} $ in ${\rm \alpha} = {\rm \nu} _n + (2{\rm \pi} nN({\rm \gamma} {\kern 1pt} {\rm \psi} )^2 /$$1 + (k^2 /2) + ({\rm \gamma} {\rm \psi} )^2 )$; they are the parameters of the generalized Airy function S(α, β, η). The resonance frequencies read

(6)$${\rm \omega} _n = n{\rm \omega} _{\rm R} = \displaystyle{{2n{\rm \omega} _0 {\rm \gamma} ^2} \over \matrix{1 + (k_{{\rm eff}}^2 /2) + ({\rm \gamma} {\rm \psi} )^2 + ({\rm \gamma} {\rm \theta} _{\rm H} )^2 \cr - {\rm \gamma} ^2 {\rm \theta} _{\rm H} {\rm \psi} \sqrt 3 (({\rm \rho} \sin {\phi} - {\rm \kappa} \cos {\phi})/{\rm \kappa} _1 )}},$$

where ω₀ = 2πc/λ_u(1 − (1/2γ²)(1 + (k ²/2))), φ is the polar angle. In the case of the common planar undulator with purely periodic magnetic field, the generalized Airy function reduces S ² → sinc ²(ν_n/2) and the only non-vanishing polarization component depends on the common Bessel function: T _n,x → (J _n+1/2( − ξ/8) + J _n−1/2( − ξ/8)), ξ = −k ²/4/(1 + (k ²/2)(1 + (d/h)²)). The conditions when the usage of the above common formulae are possible and the necessity of compensation of non-periodic magnetic components was discussed, for example, in Mikhailin et al. (Reference Mikhailin, Zhukovsky and Kudiukova2014), Zhukovsky (Reference Zhukovsky2014a, Reference Zhukovskyb, Reference Zhukovskyc). For a common planar undulator it is enough to assume d = 0 in (4)–(5) and in all following results.

It should be kept in mind that the electrons in the beam have different from each other energies, velocities, and positions respectively to the axis, which results in deviation off the axis in the angles Θ_x,y. The description of the losses due to the above mentioned factors and their effect on the UR lines shape and intensity can be done with the help of the convolution, as proposed, for example, by Dattoli (Reference Dattoli1993):

(7)$$\eqalign{& I = \int\!\!\int\!\!\int {d{\rm \varepsilon} d{\kern 1pt} \Theta _{x,y} \,(\left \vert {S({\rm \nu} _n + 4{\rm \pi} N{\rm \varepsilon} n,{\rm \beta}, {\rm \eta} )} \right \vert )^2} \cr & \quad\quad \exp [ - {\rm \varepsilon} /2{\rm \sigma} _{\rm \varepsilon} ^2 ]/(\Theta _{x,y\,{\rm max}} {\rm \sigma} _{\rm e} \sqrt {2{\rm \pi}} ),}$$

where σ_e is the energy spread, ν_n is the detuning parameter, Θ_x,ymax = ε_x,y/σ_x,y are the beam divergences, ε_x,y are the beam emittances, σ_x,y are the cross-sections, related to the emittances as follows:

(8)$$\eqalign{& {\rm \sigma} _{x,y} = \sqrt {{\rm \varepsilon} _{x,y} {\rm \beta} _{x,y}}, \quad \Theta _{x,y} = \sqrt {{{{\rm \varepsilon} _{x,y}} / {{\rm \beta} _{x,y}}}}, \quad {\rm \sigma} _{x,y} \Theta _{x,y} = {\rm \varepsilon} _{x,y},}$$

where β_x,y are the Twiss coefficients, standing for local focal lengths of the magnetic system of the undulator, chosen in an optimal way for beam matching [see Dattoli (Reference Dattoli1993) for details].

The evolution of the radiated power and of the bunching coefficients along SASE FEL depends on the fundamental Pierce parameter:

(9)$${\rm \rho} _n = \displaystyle{1 \over {2{\rm \gamma}}} \left( {\displaystyle{J \over {2{\rm \pi} i}}\left( {{\rm \lambda} _{\rm u} k_{{\rm eff}} f_n} \right)^2} \right)^{{{\rm 1} / 3}},$$

where f _n = (− 1)^n−1/2 (T _n−1 + T _n+1 + (T _n+h + T _n−h)d/h), J is the electron current density (А/m²), i = 1.704509 × 10⁴ is the Alfven current (А), n is the harmonic number. It is well known that for a stable amplification in SASE FEL the quality of the beam must be good enough and the broadening coefficients for broadening factors, such as the energy spread, divergence, etc. must be below unity, yielding the respective conditions ε_x,y ≤ λ₀/4π and σ_ε ≤ ρ/2. The Pierce parameter ρ_n renormalized as ${\tilde {\rm \rho}} _n = n^{{1 / 3}} {\rm \rho} _n $ defines for the harmonic with the number n its characteristic gain length $L_{n{\rm, g}} = {\rm \lambda} _{\rm u} /(4{\rm \pi} \sqrt 3 {\tilde {\rm \rho}} _n )$ and its maximal power $P_{n{\rm, F}} = P_{\rm F} /\sqrt n (\,f_n /nf_1 )^2 $, where $P_{\rm F} \cong \sqrt 2 {\rm \rho}_{1} P_{\rm e} $ is the power of the fundamental harmonic and P _e is the electron beam power. The SASE FEL harmonic power grows from the seeded power P _0,n and reaches the saturation at the length Z _s ≅ 1.066L _1,gln(9P _F/P ₀), following the exponential law (see Dattoli et al., Reference Dattoli, Mikhailin, Ottaviani and Zhukovsky2006a):

(10)$$P_{{\rm L,}n} (z) \cong P_{0,n} \displaystyle{{A(n,z)\exp (0.223z/Z_{\rm s} )} \over {1 + (A(n,z) - 1)P_0 /P_{\rm F}}},$$

where

(11)$$\eqalign{& A\left( {n,z} \right) \cong \displaystyle{1 \over 9}\left( {3 + 2\cosh \left( {\displaystyle{z \over {L_{n{\rm, g}}}}} \right) +} \right. \cr & \quad\quad\quad\quad \left. {4\cos \left( {\displaystyle{{\sqrt 3 z} \over {2L_{n{\rm, g}}}}} \right)\cosh \left( {\displaystyle{z \over {2L_{n{\rm, g}}}}} \right)} \right).}$$

For a prebunched beam the formula modifies accordingly to describe immediate growth without lethargy domain from the power, provided by the bunching. The radiation frequencies of a planar undulator (1) are at even harmonics with n = 1, 3, 5,…. The fundamental harmonic gives the most FEL power, the FEL subharmonics are weaker and their intensities depend as on the respective Bessel function values as on the radiation of the fundamental harmonic, being the product of the whole complex of phenomena of the interaction of the radiation emitted at resonant frequencies ω_n=1,3,5,… with the electron bunch. The sub harmonics power growth is well known to occur first following the main FEL tone at a lower power and rate; then the nonlinear regime kicks in, and the nth harmonic grows faster, following the nth power of the fundamental: Q _n(z) ∝ e ^{nz/L _g = L _1,g}. This process ends with the saturation regime on the length Z _s,n, close to that of the fundamental Z _s. The radiation power of the subharmonics of the fundamental, P _n = P _L,n + Q _n, thus consists of the so-called linear component (10), (11) and of the non-linear component as described by Dattoli et al. (Reference Dattoli, Ottaviani and Pagnutti2005b, Reference Dattoli, Mikhailin, Ottaviani and Zhukovsky2006a):

(12)$$Q_n (z) \cong P_{n,0} \displaystyle{{\exp (nz/L_{\rm g} )} \over {1 + (\exp (nz/L_{\rm g} ) - 1)P_{n,0} /P_{n{\rm, F}}}}.$$

The above nonlinear component mostly grows at the end of the undulator as the nth power of the fundamental harmonic power; it begins with the equivalent seeding power $P_{n,0} = d_n b_n^2 P_{n{\rm, F}}$ and reaches its saturated power $P_{n{\rm, F}} \cong (1/$ $\sqrt n)\,(f_n /nf_1 )^2 P_{\rm F} $, where b _n are the bunching coefficients and d ₃ ≈ 8, d ₅ ≈ 116 are the engineering constants for n = 3,5.

3. ACCOUNT FOR THE ELECTRON BEAM QUALITY

Exact account for the losses in the undulator is possible with the help of the convolution (7). Take note, that high harmonics are much more sensitive to the beam quality and are much more subjected to losses due to the dependence on n in (7): S(ν_n + 4πNεn). This influence was thoroughly explored in the case of the spontaneous UR in the array of works by Zhukovsky (Reference Zhukovsky2015a, Reference Zhukovskyb, Reference Zhukovskyc, Reference Zhukovsky2016a, Reference Zhukovskyb). However, due to rather complicated mathematical forms of integrals of generalized special functions, involved in (7), we do not think it is worth including this rigorous formalism in the simple phenomenological model of SASE FEL. We will limit ourselves to much simpler formalism of broadening or otherwise loss coefficients μ_i, describing the effective losses due to the factor i, in analogy with that done for the spontaneous UR [see, e.g., Zhukovsky (Reference Zhukovsky2014a, Reference Zhukovskyc)]; this approach seems more adequate the level of the complexity and the precision of our phenomenological model. Perhaps, the major source of losses still remains the electron energy spread, which can seriously deteriorate and even ruin the FEL operation. In particular, the energy spread increases the gain length L _g and reduces the FEL power P _F. To keep this negative effect low, FEL should be fed with highly monoenergetic beam, where σ_ε ≤ ρ/2. Importantly, following the convolution (7), we now include in the nth harmonic energy spread coefficient the dependence on the harmonic number n:

(13)$${\rm \mu} _{{\rm \varepsilon}, n} = {{2n{\rm \sigma} _{\rm \varepsilon}} / {\left( {n^{{1 / 3}} {\rm \rho} _n} \right)}}.$$

This increases the gain length for the harmonics dependently on their number:

(14)$$L_{n{\rm, g}} \to L_{n{\rm, g}} \Phi \left( {{\rm \mu} _{{\rm \varepsilon}, n}} \right)$$

and reduces their saturated power:

(15)$$P_{n{\rm, F}} \to P_{n{\rm, F}} {\rm \eta} \left( {{\rm \mu} _{{\rm e,}n}} \right)$$

with the help of the ad hoc chosen functions

(16)$$\Phi \left( {{\rm \mu} _{{\rm \varepsilon}, n}} \right) = \left( {{\rm \zeta} + 0.18{\rm \mu} _{{\rm \varepsilon}, n}^{\rm 2}} \right)\exp \left( {0.034{\rm \mu} _{{\rm \varepsilon}, n}^2} \right),\quad {\rm \zeta} = 1,$$

and

(17)$$\eqalign{ {\rm \eta} ({\rm \mu} _{{\rm \varepsilon}, n} {\rm )} &= \left( {\exp \left( { - \left( {\Phi \left( {{\rm \mu} _{{\rm \varepsilon}, n}} \right) - v} \right)\Phi \left( {{\rm \mu} _{{\rm \varepsilon}, n}} \right)} \right) + \sqrt 2 \displaystyle{{\Phi ({\rm \mu} _{{\rm \varepsilon}, n} ) - v} \over {v(\Phi ({\rm \mu} _{{\rm \varepsilon}, n} ))^3}}} \right) /w,\cr & \!\!v = 0.9,\quad w = 1.062.}$$

The eminent divergence of the beam is effectively accounted for by the coefficient ζ in (16), which depends on the proper coefficients μ_x,y, which in turn can increase the value of ζ up to ≈1.2 in the case of significant divergence, thus extending the gain and the saturation lengths. Proper relation for ζ(μ_x,y) can be derived from expressions found in FEL handbooks, such as Dattoli et al. (Reference Dattoli, Ottaviani and Pagnutti2007), and they do not qualitatively change the behavior of (16), but they make it more cumbersome; we omit them for brevity. For a well-matched good quality beams this contribution is minor and we can consider ζ = 1−1.1. Other parameters in (16) were chosen based on the comparison with the experimental data and with the results of numerical simulations of FEL dynamics in С++, available in Dattoli et al. (Reference Dattoli, Giannessi, Ottaviani and Ronsivalle2004). In particular, for the power correction we have chosen v = 0.9, w = 1.062 [see Fig. 1. Function η(μ_ε,n) – blue line for v = 0.9, w = 1.062 [see (17)]. Previously used function – red line for v = w = 1].

Fig. 1.

Function η(μ_ε,n) – blue line for v = 0.9, w = 1.062 (see. (17)). Previously used function – red line for v = w = 1.

This eliminates unphysical behavior of the previously used by Dattoli et al. (Reference Dattoli, Giannessi, Ottaviani and Ronsivalle2004) function with v = w = 1, which resulted in growth of the saturated power when μ_ε is close to, but not equal zero, and for respective energy spread σ_ε ≠ 0. Moreover, our new corrections (13)–(17) now differentiate the harmonics and have stronger effect on high FEL harmonics as it generally speaking should be according to (7).

The diffraction for the finite sized electron beam can be accounted for on the base of the beam divergence, emittance, the undulator period λ_u, and the wavelength λ_n, by modifying the Pierce parameter for the proper harmonic n as proposed, for example, in the following work of Dattoli et al. (Reference Dattoli, Ottaviani and Pagnutti2005b):

(18)$${\rm \rho} _n \to {\rm \rho} _{{\rm D,}n} = \displaystyle{{{\rm \rho} _n} \over {(1 + {\rm \mu} _{{\rm D,}n} )^{{1 / 3}}}}, \quad {\rm \mu} _{{\rm D,}n} = \displaystyle{{{\rm \lambda} _{\rm u} {\rm \lambda} _n} \over {(4{\rm \pi} )^2 {\rm \rho} _n {\rm \sigma}}}, $$

where σ = P _e/πE _eγj is the beam cross section, related to the beam power P _e and current density j, where E _e = 0.5109989 × 10⁶ is the electron energy (eV), γ is the relativistic factor, j = I ₀/(πσ) is the current density in a Gauss beam (A/m²), I ₀ is the beam current (A). More elaborated account can be done separately for x and y-coordinates, explicitly involving Twiss parameters as described by Dattoli et al. (Reference Dattoli, Giannessi, Ottaviani and Ronsivalle2004)) and making more complicated the dependence (18), but the essence of the latter relation remains unchanged. In the framework of our phenomenological model this seems just an excessive complication and formula (18) appears sufficient. In what follows we will demonstrate it by comparison with an experiment and with numerical simulations. So, corrected bunching coefficient now becomes:

(19)$$b_n = \left( {{{P_0} / {9P_{\rm e} {\rm \rho} _{{\rm D,1}}}}} \right)^{n/2}.$$

Not going deep into the details, we mention that the whole complex of the processes in SASE FEL essentially includes the interaction of the intense electromagnetic fields of the radiated frequencies with the electrons of the beam. At a certain point the radiated FEL power reaches its saturation; this process is accompanied by significant FEL induced energy spread of the electrons in the beam, increasing along the FEL length and usually exceeding by far the initial energy spread of the electron source. The FEL induced energy spread is relevant to the wave-electron beam interaction close to the saturation regime, differently from the initial energy spread, which can be rather low just out of a LINAC. The full energy spread is described by the well-known formula

(20)$${\rm \sigma} _{\rm \varepsilon} ^2 (z) = {\rm \sigma} _{{\rm \varepsilon}, 0}^2 + {\rm \sigma} _{{\rm FEL}}^2 (z),$$

where σ_ε,0 is the initial energy spread in the beam and [see, e.g., Dattoli et al. (Reference Dattoli, Ottaviani and Pagnutti2007)] the FEL induced energy spread reads as follows

(21)$${\rm \sigma} _{{\rm FEL}}^2 (z) = \displaystyle{3 \over 2}\left. {\sqrt {\displaystyle{{{\rm \rho} _{{\rm D},1} P_0 A(1,z)} \over {P_{\rm e} (1 + {{1.24P_0 (A(1,z) - 1)} / {P_{1,{\rm F}} {\rm \eta} ({\rm \mu} _{{\rm \varepsilon}, 1} {\rm )}}}}}}} \right \vert _{{\rm \rho} \to {\rm \rho} _{{\rm D,1}}},$$

where ρ → ρ_D,1 is intended everywhere in A, P _1,F, η, etc.

4. VALIDATION OF THE PHENOMENOLOGICAL FEL MODEL

Earlier it was shown that main equations, used in the phenomenological model, agree fairly well with some numerical simulations Dattoli (Reference Dattoli1993), Dattoli et al. (Reference Dattoli, Ottaviani and Renieri2005a, Reference Dattoli, Ottaviani and Pagnuttib, Reference Dattoli, Mezi, Ottaviani and Pagnutti2006b). However, no direct comparison with an experiment has been done. With the improvements and verifications introduced in the present work, in particular those, regarding the high-harmonic power and the account for the losses, we will now validate our phenomenological model by comparing it with the SASE FEL experiment, described by Giannessi et al. (Reference Giannessi, Alesini, Antici, Bacci, Bellaveglia, Boni, Boscolo, Briquez, Castellano, Catani, Chiadroni, Cianchi, Ciocci, Clozza, Couprie, Cultrera, Dattoli, Del Franco, Dipace, Di Pirro, Doria, Drago, Fawley, Ferrario, Ficcadenti, Filippetto, Frassetto, Freund, Fusco, Gallerano, Gallo, Gatti, Ghigo, Giovenale, Marinelli, Labat, Marchetti, Marcus, Marrelli, Mattioli, Migliorati, Moreno, Mostacci, Orlandi, Pace, Palumbo, Petralia, Petrarca, Petrillo, Poletto, Quattromini, Rau, Reiche, Ronsivalle, Rosenzweig, Rossi, Rossi Albertini, Sabia, Serafini, Serluca, Spassovsky, Spataro, Surrenti, Vaccarezza, Vescovi and Vicario2011). The comparison is demonstrated in Figure 2. It comes without saying that we chose the parameters for the simulation exactly matching those in the experiment; most important of them are reported below together with the results of the simulation:

• k = 2.133, h = 3, d = 0, λ_u = 0.028, N _s = IntegerPart[L _s/λ_u] = 599, L _s = 16.5 m, L _gain = 0.8265 m,

• γ = 301.37, P _E = 8.624 GW, J = 1.9 × 10⁸ A/m², Σ_ebeam = 9.38 × 10⁻⁸ m², I ₀ = 56 A, σ_ε = 0.0009, λ₁ = 504.799 nm, λ₃ = 168.266 nm, λ₅ = 100.96 nm, ρ_n = {0.0020973, 0.00116213, 0.000834944},

• ρ^D_n = {0.00185089, 0.0010721, 0.000779581},

• Φ_n = {1.18916, 4.47213, 39.1083},

• η_n = {0.922068, 0.0590925, 0.000945145},

• P _F = 20.8146 × 10⁶ W,

• P _n,F = {20.8146 × 10⁶, 14558.4, 24.0808} W,

• b _n = {0.0000834322, 5.80767 × 10⁻¹³, 4.04268 × 10⁻²¹}.

Fig. 2.

Simulation of FEL dynamics in phenomenological SASE FEL model: continuous lines, marked by arrows, n = 1 – red line, n = 3 – green line; numerical simulations results (as reported in Giannessi et al. (Reference Giannessi, Alesini, Antici, Bacci, Bellaveglia, Boni, Boscolo, Briquez, Castellano, Catani, Chiadroni, Cianchi, Ciocci, Clozza, Couprie, Cultrera, Dattoli, Del Franco, Dipace, Di Pirro, Doria, Drago, Fawley, Ferrario, Ficcadenti, Filippetto, Frassetto, Freund, Fusco, Gallerano, Gallo, Gatti, Ghigo, Giovenale, Marinelli, Labat, Marchetti, Marcus, Marrelli, Mattioli, Migliorati, Moreno, Mostacci, Orlandi, Pace, Palumbo, Petralia, Petrarca, Petrillo, Poletto, Quattromini, Rau, Reiche, Ronsivalle, Rosenzweig, Rossi, Rossi Albertini, Sabia, Serafini, Serluca, Spassovsky, Spataro, Surrenti, Vaccarezza, Vescovi and Vicario2011)): n = 1 – continuous lines, n = 3 – dashed lines; experimental data: n = 1 – circles, n = 3 – triangles.

From (1) we obtain the radiation of the fundamental harmonic at the wavelength λ₀ ≈ 500 nm and the saturation wavelength L _s = 16.5 m in agreement with that stated by Giannessi et al. (Reference Giannessi, Alesini, Antici, Bacci, Bellaveglia, Boni, Boscolo, Briquez, Castellano, Catani, Chiadroni, Cianchi, Ciocci, Clozza, Couprie, Cultrera, Dattoli, Del Franco, Dipace, Di Pirro, Doria, Drago, Fawley, Ferrario, Ficcadenti, Filippetto, Frassetto, Freund, Fusco, Gallerano, Gallo, Gatti, Ghigo, Giovenale, Marinelli, Labat, Marchetti, Marcus, Marrelli, Mattioli, Migliorati, Moreno, Mostacci, Orlandi, Pace, Palumbo, Petralia, Petrarca, Petrillo, Poletto, Quattromini, Rau, Reiche, Ronsivalle, Rosenzweig, Rossi, Rossi Albertini, Sabia, Serafini, Serluca, Spassovsky, Spataro, Surrenti, Vaccarezza, Vescovi and Vicario2011) and put by the authors as “at 15 m FEL was close to the saturation”.

It follows from Figure 2 that the phenomenological SASE FEL model not only qualitatively describes the experiment, but it also correctly reproduces the power growth, the saturated power and saturated length values and it also gives correct value for the third harmonic evolution and its saturated power. Moreover, as seen in Figure 2 our account for the losses, which is now harmonic number dependent, gives much better agreement with the experiment that most of the numerical simulations, even those in three-dimensional (3D) in C++, all shown in Figure 2 for comparison. Indeed, the dynamics of the third harmonic is such, that its growth over P ₀ = 1 W seed power of the fundamental harmonic occurs after 12 m, and it is due to the nonlinear power growth mentioned above. Most of the numerical simulations predicted much earlier growth seemingly in the linear regime (see dash-dotted lines at lengths <12 m), which, however, was not detected. It should be mentioned, that if we did not introduce harmonic dependent loss coefficient (13) and consequent correction (16), (17), sticking to the commonly used formulae, given in handbooks (example.g., Dattoli et al. (Reference Dattoli, Ottaviani and Pagnutti2007)), and then the predicted values for the third harmonic in particular would be much higher than those measured. We omit proper figure since it disagrees with the experiment.

So, our phenomenological model with harmonic dependent loss coefficient (13) and consequent corrections (16), (17) predicts correct saturated value at 15 m and rapid power growth in nonlinear regime just before 15 m, which totally agrees with the experiment. Thus, we have got good account for the beam properties. Previously used by Dattoli et al. (Reference Dattoli, Ottaviani and Pagnutti2005b, Reference Dattoli, Ottaviani and Pagnutti2007), formulae give too high values for the third harmonic power, too early growth in linear regime as results also from all numerical simulations reported in Figure 2 (see dash-dotted lines at lengths <12 m). We do not speculate on possible reasons for that, but underline that the simple phenomenological model is fully capable of giving correct predictions, which are sometimes even better than complicated numerical 3D simulations. As for the disagreement with the measurements at the first 2 m interval, we implied in the model the seed and initial lethargy regime, while in the experiment, possibly, fairly big amount of the initial shot noise was present, which led to this disagreement. Proper account for it will be introduced elsewhere.

It should be underlined that FEL dynamics is a product of the whole complex of the processes of interaction of the radiation with the electron beam, which includes the electron beam energy modulation, bunching, incoherent, and coherent radiation, bunching on the sub harmonic wavelengths and their radiation and eventually the saturation. Clearly, complete and precise description of these processes cannot be done in the framework of a simple phenomenological model fitted with very few parameters. However, the comparison with the experiment reveals surprisingly good agreement, which means that the model catches well the main features of the FEL dynamics even for high harmonics and surprisingly excels even when compared with very complex numerical simulations requiring significant computation facilities.

5. SASE FEL WITH TWO-FREQUENCY UNDULATOR FOR ROENTGEN BAND

We can now apply the validated phenomenological SASE FEL model for the simulation of the performance of SASE FEL in Roentgen range. It is well known that Roentgen laser requires high-energy electron sources and significant size of the installation. This latter increases the building and exploitation costs; moreover, on a long distance the electron bunch and the photon radiation change their positions with respect to each other, the beam divergence cannot  be neglected so that the beam should be regularly refocused; the electron energy spread also grows. Thus, it would be advantageous to obtain high-frequency radiation in an installation of a moderate scale, reducing all the above mentioned drawbacks of the present large devices. Keeping this in mind, we will model the performance of a FEL with the two-frequency undulator in order to obtain soft Roentgen radiation. We can even choose the undulator with the same parameters as used for 500 nm FEL experiments Giannessi et al. (Reference Giannessi, Alesini, Antici, Bacci, Bellaveglia, Boni, Boscolo, Briquez, Castellano, Catani, Chiadroni, Cianchi, Ciocci, Clozza, Couprie, Cultrera, Dattoli, Del Franco, Dipace, Di Pirro, Doria, Drago, Fawley, Ferrario, Ficcadenti, Filippetto, Frassetto, Freund, Fusco, Gallerano, Gallo, Gatti, Ghigo, Giovenale, Marinelli, Labat, Marchetti, Marcus, Marrelli, Mattioli, Migliorati, Moreno, Mostacci, Orlandi, Pace, Palumbo, Petralia, Petrarca, Petrillo, Poletto, Quattromini, Rau, Reiche, Ronsivalle, Rosenzweig, Rossi, Rossi Albertini, Sabia, Serafini, Serluca, Spassovsky, Spataro, Surrenti, Vaccarezza, Vescovi and Vicario2011), but with the additional second periodic field [see (3)], so that its parameters are as follows: k = 2.133, h = 3, d = −0.5, λ_u = 0.028 m and high-quality electron beam of the energy and the energy spread respectively E = 0.77 GeV, σ_ε = 0.0005; other parameters are as follows: γ = 1506.85, P _E = 103.95 GW, J = 9.96573 × 10⁹ A/m², Σ_ebeam = 4.31362 × 10⁻⁹ m², I ₀ = 135 A. With this input we evidently obtain (see (1)) the harmonics at the wavelengths λ₁ = 20.5816 nm, λ₃ = 6.86053 nm, λ₅ = 4.11632 nm, the latter being in the near-Roentgen range. The gain length, the saturation length and the total number of the undulator periods in all sections then read respectively L _gain = 1.11096 m, L _s = 25.1423 m, N _s = 898.

SASE FEL power dynamics simulation with the above given parameters for a common and for two-frequency planar undulator in the framework of our phenomenological model are presented in Figure 3. Continuous lines stand for the two-frequency undulator (left plot): n = 1 – red line, n = 3 – green line, n = 5 – blue line. For comparison we give the simulation of the performance for the common planar undulator, that is with d = 0 and with the same value of the undulator parameter k = 2.133 [см. (3)]; the electron beam also remains unchanged in this case. As seen in Figure 3 the significant power growth occurs over the 1 W seed power for the fundamental harmonic: 10⁸ times, and for the sub-harmonics: 3 × 10⁶ times for n = 3 and 10⁵ for n = 5 at the wavelength of λ₅ = 4.11632 nm, which is in the near Roentgen range. Note from the comparison of the plots in Figure 3 (left plot is for the two-frequency undulator and right plot is for the common undulator) that the high-harmonic power starts growing much earlier for the two-frequency undulator and its growth occurs much stronger and faster.

Fig. 3.

Evolution of the SASE FEL harmonic power with the two-frequency planar undulator with с k = 2.133, h = 3, d = −0.5 (left) and with the common planar undulator with d = 0 (right). The harmonic frequencies for the two-frequency undulator are λ₁ = 20.5816 nm, λ₃ = 6.86053 nm, λ₅ = 4.11632 nm (left) and for the common undulator: λ₁ = 20.192 nm, λ₃ = 6.73066 nm, λ₅ = 4.03839 nm (right). The values of the parameters of the undulator and of the beam: λ_u = 0.028 m, γ = 1506.85, P _E = 103.95 GW, J = 9.96573 × 10⁹ A/m², I ₀ = 135 A, Σ_ebeam = 4.31362 × 10⁻⁹ m², σ_ε = 0.0005. The harmonics are denoted as follows: n = 1 – red line, n = 3 – green line, n = 5 – blue line.

Moreover, the nonlinear contribution is less significant for the two-frequency undulator, big part of the harmonic power growth is due to the linear contribution at the beginning and in the middle of the undulator. We remind that this growth, being far from the saturation length is not accompanied by significant induced energy spread. In the common undulator, on the contrary, the high-harmonics grow mostly due to the nonlinear contribution, which is seen in Figure 3 as the rapid power increase at the end of the undulator close to the saturation length. Very important is the fact that the saturated power for the third harmonic in the two-frequency undulator is almost one order of magnitude higher than that for a common undulator and for the 5th harmonic it is almost two (!) orders of magnitude higher for the two-frequency undulator. This is a strong advantage of the two-frequency undulator FEL since it allows the users to get intense Roentgen radiation ≈100 kW on ≈4 nm after ≈22 m length from a relatively low-energy E ≈ 0.75 GeV beam with low seed power and frequency.

For comparison we simulate the fundamental harmonic generation at the wavelength ≈4 nm in FEL with the undulator, having the same parameters as considered before, that is with k = 2.133, λ_u = 0.028 m in Figure 4. Note, that the total length in this case equals L _s = 67.2624 m, which implies the total of N = 2402 periods (!), while the gain length becomes as long as L _gain = 3.0431 m, and the electron beam should have the gamma factor γ = 3315.07, the power P _E = 207.9 GW, the current density J = 9.96573 × 10⁹ A/m², the beam section is Σ_ebeam = 3.92147 × 10⁻⁹ m², the current I ₀ = 122.7 A; for the energy spread σ_e = 0.0005 the fundamental frequency with the wavelength λ₁ = 4.17189 nm has maximum power P _1,F = 1.11979 × 10⁸ W.

Fig. 4.

Evolution of first three odd harmonics along the SASE FEL length (in meters) for the SASE FEL with k = 2.133, λ_u = 0.028, L _s = 67.2624 m, N = 2402, L _gain = 3.0431 m, γ = 3315.07, P _E = 207.9 GW, J = 9.96573 × 10⁹ A/m², Σ_ebeam = 3.92147 × 10⁻⁹ m², I ₀ = 122.73 A, ζ = 1.1, σ_e = 0.0005, λ₁ = 4.17189 nm, λ₃ = 1.39063 nm.

Note also that the fifth harmonic is practically absent (see Fig. 4), the third harmonic at λ₃ = 1.39063 nm appears late; it is totally induced by the fundamental one and it is relatively weak at saturation level P _3,F = 10 kW after 60 m. The saturation of the fundamental harmonic, however, comes after 67 m at high power ≈100 MW. Thus, in order to generate the 4 nm Roentgen wavelength by the fundamental harmonic, the FEL installation of roughly 100 m is needed. It is technically cumbersome and complicated problem, not to mention the costs of the building and the operation. Evidently, effective high-harmonic generation, in particular in linear regime, efficiently executed by the two-frequency undulator, reduces dramatically the costs and the total scale of the FEL at the expense of lower saturated power. However, this latter shortcoming can be overcome, if advanced cascade FEL schemes are used. We will demonstrate it in forthcoming publications.

Now let us consider the simulation of the FEL with the same as before two-frequency undulator and electron beam with very low spread σ_ε = 0.0001, which is still possible for the energy E = 0.75 GeV. The evolution of the harmonic power in the framework of the phenomenological FEL model is presented in Figure 5.

Fig. 5.

SASE FEL harmonic power evolution with the two-frequency undulator: n = 1 – red line, n = 3 – green line, n = 5 – blue line. The parameters are as follows: k = 2.133, h = 3, d = −0.5, λ_u = 0.028 m, γ = 1506.85, P _E = 103.95 GW, J = 9.96573 × 10⁹ A/m², Σ_ebeam = 4.31362 × 10⁻⁹ m², λ₁ = 20.5816 nm, λ₃ = 6.86053 nm, λ₅ = 4.11632 nm, σ_ε = 0.0001.

Taken such high-quality beam we obtain even shorter gain length L _gain = 0.997478 m and saturation length L _s = 22.6088 m. Figure 5 evidences that the harmonic power in the two-frequency undulator becomes significant, ~1 and ~10 MW for the fifth and the third harmonics, respectively. The power growth is about 6–7 orders with respect to the low seed power at the fundamental frequency, the latter being n times lower than the harmonic frequency. The evolution of the power of high harmonics develops practically parallel to the fundamental harmonic power growth and it occurs in linear regime. Such behavior is due to the fact that in two-frequency undulator the Bessel functions for high harmonics have values comparable with the Bessel function values for the fundamental harmonic (see Zhukovsky, Reference Zhukovsky2015a, Reference Zhukovskyb, Reference Zhukovskyc, Reference Zhukovsky2016a, Reference Zhukovskyb), provided the losses are low as assumed. The domain of nonlinear generation, which is distinguished for the rapid growth of high harmonics at the end of the FEL undulator, reduces in the case of the two-frequency undulator. The power of the fundamental harmonic is practically the same as in common undulator: P _1,F ~ 10⁸ W, but for the 5th harmonic we obtain up to 10⁶ times growth over the seed of the fundamental harmonic, which is almost 100 times more than in a common undulator. In the considered case, we have this amplification at the Roentgen wavelength λ₅ ≈ 4 nm at just ~20 m in SASE FEL, which is times shorter than the FEL length ~70 m required to saturate the fundamental harmonic with such frequency.

The evolution of the bunching coefficients $b_n^2 $ in the above considered undulator is simulated in Figure 6, the parameters remain the same as in Figure 5.

Fig. 6.

Evolution of the bunching coefficients $b_{\rm n}^2 (z)$ for SASE FEL with two-frequency undulator: n = 1 – red line, n = 3 – green line, n = 5 – blue line. The values of the parameters are the same as given in Figure 5.

As well as for the power evolution in Figure 5, in the case of the highly monoenergetic beam with σ_ε = 0.0001 the bunching for high harmonics follows the evolution of the bunching for the fundamental harmonic in linear regime as seen in Figure 6 and the lasing effect for high harmonics in big part occurs due to the bunching on the harmonic wavelengths λ_n = λ₀/n and not only due to the bunching on the fundamental wavelength λ₀ as it happens for common undulator. In the case of a common planar undulator, the bunching coefficients for high harmonics grow slowly until nonlinear regime kicks in after 15 m. Proper figure for a common planar undulator is omitted for the sake of conciseness.

The evolution of the energy spread in FEL is essential, for example, for constructing cascade undulators. In Figure 7, we present the evolution of the energy spread along the two-frequency undulator in SASE FEL in the case of the high-quality electron beam (same beam as in Figs 5 and 6) with low energy spread σ_ε = 0.0001. As follows from Figure 7 our FEL induces the energy spread, such that at the critical length z = 18.2626 m the total energy spread reaches the maximum value, determined by the requirement σ_ε ≤ ρ/2 for the fifth harmonic; in our case it is σ_ε = 0.000484926. Indeed, for the Pierce parameter even with account for the diffraction correction we have ρ^D_n = {0.00129573, 0.00112146, 0.000973682} for the harmonics with n = 1, 3, 5. The corresponding values of the bunching read as follows:

(22)$$\eqalign{z &= 18.2626\,{\rm m},\cr\quad b_{n=1,3,5}^2 &= \left\{ {0.0678662,\;0.0144286,\;0.00116554} \right\}.}$$

At the same time the squared value of the bunching for the third and for the fifth harmonics, induced by the fundamental harmonic, read: |b _n|² ≅ c _n|b ₁|²ⁿ, where c ₁ = 1,  c ₃ = 6,  c ₅ = 60; this yields for b ²₁ = 0.0678662 [see (22)] the following values:

(23)$$b_1^2 = 0.0678662,\quad b_{3,5{\rm induced}}^2 = \left\{ {0.00187548,0.0000863811} \right\}.$$

Upon the comparison of (22) and (23) we conclude that the induced by the fundamental harmonic values (23) are almost one order less than those due to the independent growth (22) and in the two-frequency undulator we have approximately one order of magnitude stronger bunching of high harmonics than in the common undulator; in the latter it is largely determined by power growth of the fundamental harmonic. This fact is important for building a cascade FEL, in which the last section is tuned on the harmonic of the preceding sections. Owing to this fact the two-frequency undulator appears particularly appealing solution in the cascade high-gain harmonic generation SASE FEL. Indeed, the last amplifer–radiator section, tuned at the third or at the fifth harmonic of the two-frequency undulator prebuncher section, gets already quite strong bunching [see (22)] at short length z = 18.2 m, in particular as compared with that in the case of a common planar undulator. Moreover, the amplifier-radiator gets the electron beam with low induced energy spread. Thus the beam, prebunched in the two-frequency undulator at the third or the fifth harmonic wavelength, can be efficiently used for further amplification of the proper harmonic in the amplifier section. The details will be addressed in a forthcoming publication.

Fig. 7.

Evolution of the energy spread in SASE FEL with two-frequency undulator.