Effect of the third undulator field harmonic on spontaneous and stimulated undulator radiation

Zhukovsky, K.V.

doi:10.1107/S1600577519008415

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 26| Part 5| September 2019| Pages 1481-1488

https://doi.org/10.1107/S1600577519008415

Effect of the third undulator field harmonic on spontaneous and stimulated undulator radiation

K. V. Zhukovsky ^a ^*

^aDepartment of Theoretical Physics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119991, Russian Federation
^*Correspondence e-mail: zhukovsk@physics.msu.ru

Edited by M. Yamamoto, RIKEN SPring-8 Center, Japan (Received 3 January 2019; accepted 13 June 2019; online 15 August 2019)

The effect of undulator field harmonics on spontaneous and stimulated undulator radiation, both on and off the undulator axis, is studied. Bessel factors for the undulators with field harmonics have been analytically calculated and numerically verified. The influence of the third undulator field harmonic on single-pass free-electron laser radiation is explored. Harmonic generation at the LCLS and SPring-8 free-electron lasers is modeled and analyzed.

Keywords: undulator radiation; harmonics; free-electron laser.

1. Introduction

In an undulator, electrons with high relativistic factor γ pass through a spatially periodic magnetic field and emit undulator radiation (UR). Undulators in free-electron lasers (FELs) generate coherent UR (McNeil & Thompson, 2010 ; Pellegrini et al., 2016 ; Huang & Kim, 2007 ; Saldin et al., 2000 ; Margaritondo & Ribic, 2011 ; Pellegrini, 2016 ). This process can be modeled numerically or using phenomenological formulae. The former approach implies a numerical solution of an equation system for the electron motion and interaction with the wavefield. This can be done in one or three dimensions and requires serious computational resources and trained personnel. The analytical approach gives an effective FEL description and can reproduce fairly well the harmonic growth in real devices. Going beyond the usual assumption of a sinusoidal undulator on-axis field, we study the following undulator field,

$[{\bf{H}} = \left( 0,H_0 \big\{ 0+\big[\sin\left(zk_\lambda\right)+d\sin\left(hzk_\lambda\right)\big] \big\}, 0 \right), \eqno(1)]$

with amplitude H₀, main period λ_u, k_λ = 2π/λ_u, second period λ_u/h and second amplitude dH₀, h = 2, 3, 4…. A relatively weak, h ≃ 0.1–0.5, third field harmonic can be present in real devices. There have been controversial conclusions (Zhukovsky, 2015a ,b , 2016a ,b ; Jeevakhan & Mishra, 2011 ; Mishra et al., 2009 ; Jia, 2011 ) on its effect on the UR harmonics. We rigorously obtained the analytical expressions for the Bessel functions T_n for the undulator field (1) by computing the radiation integral

$[\eqalignno{ {{{\rm{d}}^2I}\over{{\rm{d}}\omega\,{\rm{d}}\Omega}} = {}& {{e^{2}}\over{4\pi^{2}c}} \Bigg| \,\omega\!\!\int\limits_{-\infty}^{\infty} \big[{\bf{n}}\times\left({\bf{n}}\times{\boldbeta}\right)\big] \exp\left[i\omega\left(t-{\bf{n}}{\bf{r}}/c\right)\right] \,{\rm{d}}t \Bigg|^2, &(2) }]$

where

$[{\bf{n}} \cong \left\{\theta\cos\phi,\theta\sin\phi,1-\theta^2/2\right\}]$

where $[{\bf{r}}]$ are coordinates and $[\boldbeta]$ are velocities.

The exponential in (2) yields the generalized Bessel function

$[\eqalignno{ J_n = {}& {{1}\over{2\pi}} \int\limits_{0}^{2\pi} \cos \Big\{ \xi_1\sin\phi + \xi\sin(2\phi) + \xi_{-}\sin\left[(h-1)\phi\right] \cr& + \xi_{+}\sin\left[(h+1)\phi\right] + \xi_{h}\sin(2h\phi)+n\phi \Big\} \,{\rm{d}}\phi, & (3) }]$

which involves the parameter k = $[H_0\lambda_{\rm{u}}e/2{\pi}mc^2]$ as follows,

$[\eqalign{ \xi & = {{nk^2/4}\over{1+(k^2/2)\left[1+(d^2/h^2)\right]_{\vphantom{\big|}}}}, \cr \xi_1 & = {{8\xi\gamma\theta}\over{k}}, \qquad \xi_{-}={{4d\xi}\over{h(h-1)}}, \cr \xi_{+} & = {{4d\xi}\over{h(h+1)}}, \qquad \xi_{h}={{d^2\xi}\over{h^3}}. } \eqno(4)]$

The Bessel coefficients $[f_{n\semi{x}}]$ = $[|T_{n\semi{x}}|]$ for the harmonic n = 1, 2, 3,… taking account of the off-axis radiation at angle θ read as follows,

$[{T_{n,x}} = {J_{n - 1}} + {J_{n + 1}} + {d \over h}\left(\,{{J_{n + h}} + {J_{n - h}}} \right) + a{J_n},\quad {T_{n,y}} = b{J_n}, \eqno(5)]$

where a = $[2\gamma\theta\cos\phi/k]$ , b = $[2\gamma\theta\sin\phi/k]$ , θ is the off-axis angle and ϕ is the polar angle. The wavelength of the nth UR harmonic from a planar undulator is

$[\lambda_n = {{{\lambda_{\rm{u}}}} \over {2n{\gamma ^2}}} \left[{1 + k_{\rm{eff}}^2 + {{\left({\gamma\theta}\right)}^2}}\right], \quad k_{\rm{eff}}^2 = {{{k^2}}\over{2}}\left({1+{{{d^2}}\over{{h^2}}}}\right). \eqno(6)]$

In the following we present the Bessel functions also for the elliptic undulator with field harmonics. We explore in particular the effect of the third undulator field harmonic on the UR. We compute analytically the UR intensity in the field (1) and compare it with proper numerical results. Furthermore, we calculate the UR intensity for an elliptic undulator and we model the harmonic power in several FEL experiments. We analyze the harmonic generation in the SPring-8 and LCLS FELs for several setups and study possible effects of the undulator field harmonics.

2. Spontaneous UR and validation of the Bessel factors

The intensity of the spontaneous UR in the undulator with N periods taking into account the energy spread σ_e is given by the convolution

$[\eqalignno{ {{{\rm{d}}^2I_n}\over{{\rm{d}}\omega\,{\rm{d}}\Omega}} \cong {}& C\left(\left|T_{n,x}\right|^2+\left|T_{n,y}\right|^2\right) \int\limits_{-\infty}^{\infty} \exp\left(-\varepsilon^2/2\sigma_{\rm{e}}^2\right) \cr& \times {\rm{sinc}}^2\left(\nu_{\rm{n}}+2{\pi}nN\varepsilon\right) \,{\rm{d}}\varepsilon, & (7) }]$

where C = $[e^2N^2\gamma^2k^2n^2/c\sqrt{2\pi}\sigma_{\rm{e}}(1+k_{\rm{eff}}^2)^2]$ , e is the electron charge, c is the speed of light and $[\nu_n]$ = $[2{\pi}nN[(\lambda_n/\lambda)-1]]$ is the detuning parameter. On the axis of a common planar undulator, where d = θ = 0, only x-polarization is radiated. The proper Bessel coefficient f_n,x = $[J_{(n-1)/2}(-\xi_0)+J_{(n+1)/2}(-\xi_0)]$ contains common Bessel functions $[J_n(\xi_0\equiv\xi|_{d\,=\,0})]$ . In the presence of the additional field harmonic the Bessel coefficients f_n of the planar undulator (1) depend on k and d [see (3)–(5)]. An example of such a dependence for h = 3 is shown in Fig. 1. The energy spread decreases the effective value of f_n as shown in Fig. 2.

Figure 1
Bessel coefficients f_1,3,5 and their ratio f₃/f₁ for k and d, h = 3.

Figure 2
Bessel factors, weighted with the energy spread σ_e = 10⁻⁴ (left) and σ_e = 10⁻³ (right), γθ = 0.0123, k = 2.1, h = d = 0.

It follows from the analysis of Fig. 1 that the Bessel factors and the UR harmonics weakly sense the third undulator field harmonic for k ≅ 1.5. For k < 1.5 the third field harmonic with negative phase, d < 0, slightly enhances the UR harmonics. The positive phase of the field harmonic, d > 0, weakens the UR harmonics (see Fig. 1). For k > 1.5 the effect of the field dH₀ in (1) is the opposite: high UR harmonics with n = 3, 5,… become stronger for d > 0 and weaker for d < 0 (see Fig. 1). This observation clarifies contradicting reports on the harmonic behavior in various references (Zhukovsky, 2015a,b, 2016a,b; Jeevakhan & Mishra, 2011; Mishra et al., 2009; Jia, 2011). Indeed (see Fig. 1), for k > 1.5 the rate f_3,5/f₁ increases for higher values of d > 0 and decreases for lower values d < 0. For k < 1.5, on the contrary, the rate f_3,5/f₁ of high harmonics increases for lower values of d < 0. The variation of f_3,5/f₁ can be about ±20%, depending on d.

We have computed the UR harmonic intensity numerically using the SPECTRA program (Tanaka & Kitamura, 2001 ; Tanaka, 2014 ), which allows field harmonics for planar undulators. Examples for k = 2.1, h = 3, d = ±0.42, σ_e = 1.5 × 10⁻³ are shown in Figs. 3 and 4, where color bars show our analytical results and thin black lines show the data from SPECTRA. The agreement is quite good. The ratio I_n/I₁ of the spontaneous UR harmonic intensities changes, for d = ±0.42, by about ±25% for n = 1, 3, 7; this ratio almost does not change for harmonic n = 5 in our example for k = 2.1 (compare Fig. 3 with Fig. 4); this agrees with the results in Fig. 1. The power variation for high harmonics with n > 7 can be ∼300%, but they remain rather weak. Odd harmonics are very weak.

Figure 3
Spontaneous UR harmonic intensity (in relative units) from the undulator with k = 2.1, h = 3, d = +0.42, σ_e = 1.5 × 10⁻³ for the harmonic number n.

Figure 4
Spontaneous UR harmonic intensity (in relative units) from the undulator with k = 2.1, h = 3, d = −0.42, σ_e = 1.5 × 10⁻³ for the harmonic number n.

In Fig. 5 we demonstrate the shape of the periodic magnetic field with harmonics. The third undulator field harmonic with negative phase, d < 0, increases the amplitude of the magnetic field (see Fig. 5). For d > 0, the amplitude of the magnetic field may even decrease (see Fig. 5).

Figure 5
The undulator magnetic field for k = 2.1, h = 3, d = ±0.1 (left plot), d = ±0.42 (right plot), d = 0 (solid lines), d > 0 (dotted lines), d < 0 (dashed lines).

3. Phenomenological description of FEL harmonics evolution

The analytical formulation of the power evolution in a single-pass FEL was developed by Dattoli and co-workers (Dattoli & Ottaviani, 2002 ; Dattoli et al., 2004 , 2005a ,b ); it employs the logistic function. The model was improved and calibrated with FEL experiments by Zhukovsky and co-workers (Zhukovsky & Potapov, 2017 ; Zhukovsky, 2017a ,b , 2018 ; Zhukovsky & Kalitenko, 2019a ,b ). In the following we present its development, which describes high harmonic growth around the saturated region. The Pierce parameter for the nth FEL harmonic (Dattoli et al., 2005b) reads as follows,

$[\eqalign{ & {\rho_{\rm{n}}} = {{ {J^{1/3}}{{\left({{\lambda_{\rm{u}}}k\,{f_n}}\right)}^{{{\rm{2}}/3}}} }\over{ 2\gamma{{\left({4{\pi}i}\right)}^{1/3} }}}, \quad {\tilde \rho _n} = {{{\rho _n}} \over {{{\left({1 + {\mu _{D,n}}} \right)}^{{1/3}}}}}, \cr&\qquad\qquad\,\,\,\, {\mu _{D,n}} \cong {{{\lambda_{\rm{u}}}{\lambda _n}} \over {16\pi {\rho _n}\Sigma }}, } \eqno(8)]$

where accounting for diffraction comes through the beam section $[\Sigma]$ = $[2\pi({\beta_x}\varepsilon_x{\beta_y}\varepsilon_y)^{1/2}]$ for harmonic wavelength λ_n and undulator period λ_u; J is the current and f_n is the Bessel coefficient. The corrected gain length is $[{L_{n,{\rm{g}}}} \cong]$ $[{{{\Phi_n}\,{\lambda_{\rm{u}}}}/{4\pi\sqrt{3}{n^{{1/3}}}{{\tilde\rho}_n}}}]$ , the saturation occurs at $[{L_{\rm{s}}} \cong]$ $[1.07{L_{1,\,{\rm{g}}}}\ln({{9{\eta_1}{P_F}}/{{P_{1,0}}}})]$ , the saturated harmonic powers are $[{P_F} \cong]$ $[\sqrt{2}{P_{\rm{e}}}{{{\eta_1}\tilde\rho_1^{\,2}}/{{\rho _1}}}]$ , P_n,F = $[{{{\eta_n}{P_F}\,f_n^{\,2}}/{{n^{{5/2}}}\,f_1^{\,2}}}]$ and P_e is the electron beam power. The following phenomenological corrections account for the beam size and the energy spread σ_e,

$[\eta_n \cong \Big\{ \exp\Big[-\Phi_n\left(\Phi_n-0.9\right)\Big] + 1.57\left(\Phi_n-0.9\right)/\Phi_n^3\Big\}/1.062, \eqno(9)]$

$[\eqalign{ \Phi_n & \cong \left(\zeta^n+0.165\mu_{\varepsilon,n}^2\right) \exp\left(0.034\mu_{\varepsilon,n}^2\right), \cr \mu_{\varepsilon,n}(\sigma_\varepsilon) & \cong 2\sigma_\varepsilon/n^{1/3}{\tilde\rho}_n,} \eqno(10)]$

$[\zeta \cong 1+0.07\textstyle\sum\limits_i \mu_i+0.35\textstyle\sum\limits_i \mu_i^2, \eqno(11)]$

$[{\mu_{\tilde x,\tilde y}} = {1 \over {\tilde \rho }}{{{\gamma ^2}{\varepsilon _{x,y}}} \over {(1 + {{{k^2}}/2}){\lambda_{\rm{u}}}{\beta _{x,y}}}}, \quad {\mu _{x,y}} = {1 \over {\tilde \rho }}{{{\gamma ^2}\omega _\beta ^2{\varepsilon _{x,y}}} \over {(1 + {{{k^2}}/2}){\gamma _{x,y}}}}, \eqno(12)]$

where ∊_x,y are the emittances, β_x,y and γ_x,y are the Twiss parameters, $[{\omega_\beta}]$ = $[{{\pi k}/{\gamma {\lambda_{\rm{u}}}}}]$ is the betatron frequency, $[{\tilde \rho_n}]$ < $[\rho]$ by ∼15–30%, $[\zeta]$ ≃ 1–1.05. The conditions for stable amplification (McNeil & Thompson, 2010; Pellegrini et al., 2016; Huang & Kim, 2007; Saldin et al., 2000) are $[{\sigma _\varepsilon }]$ $[\le]$ $[{{{{\tilde \rho }_n}}/2}]$ , $[{\varepsilon_{x,y}}]$ $[\le]$ $[{{{\lambda_n}}/{4\pi}}]$ . The radiation power exponentially grows along the FEL; for the initially unbunched beam it reads (Dattoli et al., 2004, 2005a,b) as follows,

$[\eqalignno{ {P_{L,n}}\left(z \right) & \cong {{{P_{0,n}}\,A\left({n,z} \right)\,{\exp\left({{0.223z}/{{Z_s}}}\right)}} \over {1 + \left[{A\left({n,z} \right) - 1} \right]{{{P_{0,n}}}/{{P_{n,F}}}}}}, &(13) \cr A\left({n,z} \right) & \cong {1 \over 3} + {{\cosh ({z/{{L_{n,{\rm{g}}}}}})} \over {4.5}} + {{\cos ({{\sqrt{3}z}/{2{L_{n,{\rm{g}}}}}}) \cosh({z/{2{L_{n,{\rm{g}}}}}})} \over {0.444}}, }]$

where P_0,n is the initial power for the nth harmonic. The nonlinear power term, induced by the fundamental tone, is (Dattoli et al., 2005b)

$[{Q_n}\left(z \right) \cong {P_{n,0}}{{\exp \left({{{n\,z}/{{L_{\rm{g}}}}}} \right)} \over {1 + \left[{\exp \left({{{n\,z}/{{L_{\rm{g}}}}}} \right) - 1} \right]{{{P_{n,0}}}/{{P_{n,F}}}}}}, \eqno(14)]$

where $[{P_{n,0}} \cong b_n^2\,{P_{n,F}}]$ is the equivalent initial power due to the induced bunching b_n² $[\cong]$ $[{({{{{P_{0,1}}}/{9{P_{\rm{e}}}\,{{\tilde \rho }_1}}}})^n}]$ . The power evolution in sectioned undulators, the bunching coefficients, energy spread and other details can be found elsewhere (Dattoli et al., 2005a,b, 2017a,b, 2018; Zhukovsky & Potapov, 2017; Zhukovsky & Kalitenko, 2019a).

Equations (13) and (14) though do not describe the harmonic evolution close to saturation. In reality the harmonic power saturates gradually, while the sum of (13) and (14) describes the saturation in one step. To address multi-stage harmonic saturation we introduce another broadening coefficient, $[{\tilde \mu _{\varepsilon, n}}({\sigma _\varepsilon })]$ $[\cong]$ $[{{2n{\sigma _\varepsilon }}/{({n^{{1/3}}}{{\tilde \rho }_n})}}]$ , which involves an additional factor n to describe higher losses for high UR harmonics and their spectrum line broadening in accordance with Zhukovsky (2015a,b, 2016b). This yields other broadening coefficients,

$[\tilde\eta_n \cong \Big\{ \exp\left[-\tilde\Phi_n\left(\tilde\Phi_n-0.9\right)\right] + 1.57\left(\tilde\Phi_n-0.9\right)/\tilde\Phi_n^3 \Big\} / 1.062, \eqno(15)]$

$[\eqalign{ {\tilde \Phi _n} & \cong \left({{\zeta ^{\,n}} + 0.165\tilde \mu _{\varepsilon, n}^2} \right) \exp\left(0.034\tilde\mu_{\varepsilon,n}^2\right), \cr {\tilde \mu _{\varepsilon, n}}({\sigma _\varepsilon }) & \cong {{2{n^{{2/3}}}{\sigma _\varepsilon}}/{{{\tilde \rho }_n}}}.} \eqno(16)]$

They in turn modify the power values $[{\tilde P_{n,F}}]$ = $[{{{{\tilde \eta }_n}{{\tilde P}_F}f_n^2}/{{n^{{5/2}}}\,f_1^{\,2}}}]$ , $[{\tilde P_F}]$ $[\cong]$ $[\sqrt 2 {P_{\rm{e}}}{{{{\tilde \eta }_1}\tilde \rho _1^{\,2}}/{{\rho _1}}}]$ , $[{\tilde P_{n,0}}]$ $[\cong]$ $[{d_n}b_n^2{\tilde P_{n,F}}]$ and the gain length $[{\tilde L_{n,{\rm{g}}}}]$ $[\cong]$ $[{{{{\tilde \Phi }_n}{\lambda_{\rm{u}}}}/{4\pi \sqrt 3 {n^{{1/3}}}{{\tilde \rho }_n}}}]$ ; the coefficients d_n $[\cong]$ { 1, 3, 8, 40, 120 } describe the anticipated harmonic power growth up to the first stage of saturation. A new nonlinear term appears,

$[{\tilde Q_n}\left(z\right) \cong {\tilde P_{n,0}}\,{{\exp \left({{{n\,z}/{{L_{\rm{g}}}}}} \right)} \over {1 + \left[{\exp \left({{{n\,z}/{{L_{\rm{g}}}}}} \right) - 1} \right]{{{{\tilde P}_{n,0}}}/{{{\tilde P}_{n,F}}}}}}, \eqno(17)]$

which should be taken into account together with (14). The results of the modeling with (8)–(17) are presented in the following section. The phenomenological model is flexible: intersectional losses, imposed by harmonic filtering or phase shifters, can be introduced etc. It can be easily implemented in any computer program, such as Mathematica, and it allows instant analysis of FELs.

4. Effect of the third field harmonic on the planar FEL radiation

Using our new formulation of the total FEL power, $[{P_{L,n}} + {Q_n} + {\tilde Q_n}]$ , we modeled several well documented FEL experiments; below we present some examples, comparisons with the measurements and numerical simulations. We modeled an LCLS experiment, where the radiation at λ₁ = 0.15 nm was produced in the FEL, built with 17 undulators, each 3.4 m long, with 15 cm gaps between them, k = 3.5, λ_u = 3 cm. The electron energy was E = 13.6 GeV, the energy spread σ_e = 1 × 10⁻⁴, the emittance ∊_x,y = 0.4 µm rad [see Emma et al. (2010 ) for details]. The saturated power in our model was ≃20 GW, the gain length L_g = 3.7 m, the saturation length L_s ≃ 56.5 m of pure undulators; L_s ≃ 60 m, including the gaps. The measurements (Ratner et al., 2011 ) gave the power rate of high harmonics as 0.2 to <2% of the fundamental, which included all high harmonic contributions. The fifth harmonic power was estimated at ∼0.1 of the power of the third harmonic; the second harmonic was not registered in the experiment. Our modeling results and the experimental values agree very well.

Soft X-rays, λ₁ = 1.5 nm, were produced in the LCLS experiment for E = 4.3 GeV, I₀ = 1 kA, σ_e = 3 × 10⁻⁴, ∊_x,y = 0.4 mm mrad. In this case the measured power of the third harmonic was 2–3% of the fundamental, the fifth harmonic was estimated at ∼0.1 of the third harmonic power and the second harmonic power was <0.1%. A comparison of the results of our modeling with the experimental range is presented in Fig. 6. The harmonic powers are within or close to the experimental range, denoted by the shadowed areas on the right in Fig. 6. The effect of the third undulator field harmonic on the radiated UR harmonics is shown in Fig. 6, where we omitted the noise contribution to show pure power growth. Note the four-stage evolution of the harmonic powers, described by our model: the lethargy region <5 m is followed by the independent harmonic growth (13), after 15 m the non-linear generation (17) develops until the first saturation at ∼20 m, and then the growth still proceeds (14) until the final saturation power is reached.

Figure 6
Effect of the third undulator field harmonic on the FEL power in the low-energy LCLS experiment with E = 4.3 GeV, I₀ = 1 kA, σ_e = 3 × 10⁻⁴, ∊_x,y = 0.4 mm mrad, β = 15 m, λ_u = 3 cm, k = 3.5. Modeled power for harmonics: n = 1 (red), n = 2 (orange), n = 3 (green), n = 5 (blue); d = −0.3 (dashed lines), d = +0.3 (dotted lines). The experimental saturated power range for harmonics is denoted by shadowed areas.

For d = −0.3 (dashed lines in Fig. 6) we obtained some shorter gain and saturation lengths, a slightly stronger fundamental and slightly weaker high harmonics. For d = +0.3 (dotted lines in Fig. 6), the effect was the opposite. It was noticeable more for the gain length than for the saturated harmonic powers, although the latter changed by ∼25%. The second FEL harmonic was much weaker than the fifth, though its saturated power was higher than the initial power of the fundamental tone.

In the LEUTL FEL experiment (Milton et al., 2001 ) the radiation at λ₁ = 385 nm was produced by the current I₀ = 184 A of electrons with γ = 500. Our modeling agrees well with the experimental results: the saturation power 0.1 GW, length L_s ≃ 15 m and gain L_g = 0.8 m are all well reproduced. The harmonic powers reach saturation in two stages and stay within the experimental range. The effect of the third field harmonic is similar to that in Fig. 6 and we omit it for brevity.

Similarly good was the match with the SPARC experiment (Giannessi et al., 2011 ), where quite low energy electrons with γ = 297 emitted radiation at 0.5 µm. The phenomenological modeling yields a pure undulator FEL length L_s = 13.3 m and L_g = 0.64 m; this agrees with the measurements of Giannessi et al. (2011), with GENESIS numerical simulations by Alesini et al. (2004 ) and with our own numerical simulations. Details of the above examples are given by Zhukovsky (2019 ).

We modeled FEL experiments at the SPring-8 X-ray source in its upgraded (Owada et al., 2018 ) and original (Shintake et al., 2009 ) setups. The FEL power evolution in the SACLA FEL experiment (Owada et al., 2018) with the current I = 120 A, γ = 1570, σ_e = 3 × 10⁻⁴, ∊_x,yⁿ = 1 mm mrad, λ_u = 1.8 cm, k = 2.1 is shown in Fig. 7. We obtained a gain length L_g = 1 m and saturation length L_s = 12.7 m, as reported by Owada et al. (2018); the harmonic powers for the radiation at λ₁ = 12 nm, λ₃ = 4 nm, λ₅ = 2.4 nm are P_F,n ≃ 1 × 10⁸, 1 × 10⁶, 8 × 10⁴ W. The rate P_F,3/P_F,1 ≃ 0.9% agrees quite well with the 0.5% reported by Owada et al. (2018) as well as P_F,5/P_F,3 ≃ 7%. The effect of the third undulator field harmonic with d = ±0.3 in (1) on the FEL radiation was insignificant in this experimental setup; it has been omitted in Fig. 7.

Figure 7
FEL harmonic power evolution in the SACLA FEL experiment with I = 120 A, γ = 1570, σ_e = 3 × 10⁻⁴, ∊_x,yⁿ = 1 mm mrad, λ_u = 1.8 cm, k = 2.1. Harmonics: n = 1 (red), n = 3 (green), n = 5 (blue), n = 2 (orange dashed), n = 4 (violet dotted).

In SPring-8 installations the undulator parameter k varies over a wide range. We modeled another experimental setup (Shintake et al., 2009) with k = 0.7, λ_u = 1.5 cm, I = 300 A, γ = 489, σ_e = 2 × 10⁻⁴, ∊_x,yⁿ = 2.7 mm mrad, λ₁ = 39 nm, λ₃ = 13 nm, λ₅ = 8 nm (see the FEL power in Fig. 8).

Figure 8
FEL harmonic power evolution in the SPring-8 FEL experiment with I = 300 A, γ = 489, σ_e = 2 × 10⁻⁴, ∊_x,yⁿ = 2.7 mm mrad, λ_u = 1.5 cm, k = 0.7. Harmonics: n = 1 (red), n = 3 (green), n = 5 (blue), n = 2 (orange). d = 0 (solid lines), d = −0.3 (dashed lines), d = 0.3 (dotted lines).

In Fig. 8, solid lines describe radiated harmonics from the FEL undulator with an ideal sinusoidal field, d = 0. The additional third undulator field harmonic strongly influences the FEL harmonic powers in this setup (see Fig. 8). The third FEL harmonic power reaches 160 kW for d = −0.3, 55 kW for d = 0, and just 7 kW for d = 0.3; the variation is ∼200%. The fifth FEL harmonic is weak; it does not develop for d = 0.3 and it reaches 0.5 kW for d = −0.3. The first and second harmonics are almost not affected (see Fig. 8).

5. Effect of the third field harmonic on the elliptic FEL radiation

In some elliptic undulators the harmonics of the undulator magnetic field can be significant. Lee et al. (2015 ) studied a bi-harmonic undulator numerically; the amplitude of the main undulator field was 9.7 kG and the amplitude of the third field harmonic was 0.8 kG. The magnetic field in the undulator was

$[{\bf{H}} = {H_0}\left[\matrix{ \sin ({k_\lambda }z) - d\sin (h{k_\lambda }z) \cr \cos ({k_\lambda }z) + d\cos (h{k_\lambda }z) \cr 0} \right], \eqno(18)]$

the undulator period was λ = 2.3 cm, k = 2.21622, h = 3, d = 0.0825. The wavelength of the UR from the undulator with the field (18) is given by (6), where k_eff² = k²[1+(d²/h²)]. We have performed rigorous analytical calculations and obtained the Bessel functions for this undulator. The radiation integral for the undulator field (18) yields the following generalized Bessel functions,

$[\eqalignno{ J_n^{\,h}\left({{\xi_1},{\xi_2},{\xi_3},{\xi_4},{\xi_5}}\right) = {}& {{1}\over{2\pi}} \int\limits_{-\pi}^\pi {\rm{d}}\alpha \exp \Big( i \Big\{ n\alpha+\xi_1\cos\alpha \cr& + \xi_2\cos\left(h\alpha\right) - \xi_3\sin\alpha + \xi_4\sin\left(h\alpha\right) \cr& - \xi_5\sin\left[\left(h+1\right)\alpha\right] \Big\} \Big), & (19)}]$

$[\eqalign{ {\xi_1} & = {{2nk\gamma\theta\cos\phi}\over{1+{k^2}{{\left[{1+{{\left({d/h}\right)}^2}+{\gamma^2}{\theta^2}}\right]}}}}, \cr {\xi_2} & = {{d}\over{h^2}}\,{\xi_1}, \quad {\xi_3}={\xi_1}\tan\phi, \quad {\xi_4} = {{d}\over{h^2}}\,{\xi_1}\tan\phi, \cr {\xi_5} & ={{2ndk^2}\over{h\left(h+1\right){{\left\{{1+{k^2}\left[{1+{{\left({d/h}\right)}^2}+{\gamma^2}{\theta^2}}\right]}\right\}}}}}. } \eqno(20)]$

The UR intensity is given by (7), where k_eff² = k²[1+(d²/h²)]. The generalized Bessel functions (19) yield the following amplitudes for the spontaneous UR in the angle θ off the axis,

$[{T_{n\semi x}} = {{2\gamma } \over k}\theta \cos \phi\,\, J_n^{\,h} + i\left({J_{n + 1}^{\,h} \!- J_{n - 1}^{\,h}} \right) + i{d \over h}\left({J_{n + 1}^{\,h} \!- J_{n - h}^{\,h}} \right), \eqno(21)]$

$[{T_{n\semi y}} = {{2\gamma } \over k}\,\theta \sin \phi\,\, J_n^{\,h} - \left({J_{n + 1}^{\,h} + J_{n - 1}^{\,h}} \right) + {d \over h}\left({J_{n + h}^{\,h} + J_{n - h}^{\,h}} \right). \eqno(22)]$

The intensities of the spontaneous UR in the undulator (18) with the data taken from Lee et al. (2015) are shown in Fig. 9 for d = 0.0825 and d = 0.3 in the upper and lower plots, respectively.

Figure 9
Spontaneous UR harmonic intensity in the undulator (18)

for h = 3, d = 0.0825 (top plot) and d = 0.3 (bottom plot) (in relative units).

The second harmonic is evident in the UR spectrum; its power is 8.7% of the fundamental tone; ideally, only the first harmonic should be radiated by the helical undulator on the axis. Note also that the fifth harmonic is stronger than the third; this is due to the third undulator field harmonic with amplitude d in (18). For d = 0.3 its power is 15.5% of the fundamental (see lower plot in Fig. 9). The Bessel coefficients are $[{f_{n\semi x,y}}]$ = $[|{{T_{n\semi x,y}}}|]$ . On the undulator axis θ = 0, only the ξ₅ argument survives in $[J_n^{\,h}({{\xi _i}})]$ and we obtain (21) and (22) with $[J_n^{\,h}({{\xi_5}})]$ = $[\textstyle\int_{-\pi}^\pi {\rm{d}}\alpha \exp(i\{n\alpha-\xi_5\sin[(h+1)\alpha]\})/2\pi]$ . For d = 0 we obtain a common result for the spiral undulator: $[{f_{1\semi x,y}}]$ = 1 and $[{f_{n \ne 1}}]$ = 0.

We have applied the phenomenological FEL model to describe the evolution of the harmonic power in the FEL with the bi-harmonic helical undulator (18). We considered the beam of the LCLS installation with the low-energy value, E = 4.3 GeV, and the SACLA beam with E = 800 MeV, where the planar undulator with close value of k was employed. The modeling data are the following: γ = 8400, P_E = 4292 GW, J = 2.23 × 10¹¹ A, Σ_beam = 4.49 × 10⁻⁹ m², I₀ = 1 kA, σ_e = 0.0003, ∊ⁿ = 0.4 × 10⁻⁶ µm rad, β = 15 m, ζ = 1.05, k = 2.21622, h = 3, λ_u = 2.3 cm, L_s = 24.5 m, L_g = 1.5 m, λ₁ = 0.97 nm, λ₂ = 0.49 nm, λ₃ = 0.32 nm, λ₄ = 0.24 nm, λ₅ = 0.19 nm.

For d = 0.3,

$[\eqalign{f_{n\semi y,x} & = \{0.9958, 0.0550, 0.0380, 0.0162, 0.0984\}, \cr \rho_{n\semi y,x} & \simeq \{0.00080, 0.00011, 0.00008, 0.00005, 0.00017\}, \cr P_{n,y,x,F} & \simeq \{4.4\!\times10^9, 44.3\times10^3, 4.41\times10^3, 4.4, 211\!\times10^3\}\,{\rm{W}}. }]$

For d = 0.08247,

$[\eqalign{ f_{n\semi y,x} & = {0.9980,0.0557,0.0109,0.0047,0.0275}, \cr \rho_{n\semi y,x} & \simeq \{0.00080,0.00011,0.00003,0.00002,0.00007\}, \cr P_{n,y,x,F} & \simeq \{4.45\times10^9, 47.3\times10^3, 0, 04, 560\}\,\,{\rm{W}}. }]$

The simulations of the FEL power evolution along the undulators for d = 0.08247 and for d = 0.3 are shown in Fig. 10.

Figure 10
Harmonic power evolution in the cos–sin spiral undulator FEL with the field (18)

for γ = 1570, I₀ = 120 A, ∊ⁿ = 10⁻⁶ mm mrad, β = 0.37 m, k = 2.216, h = 3, λ_u = 2.3 cm; both x- and y-polarization have the same power. Harmonics: n = 1 (red), n = 2 (yellow), n = 3 (green), n = 4 (green dashed), n = 5 (blue dotted). Undulator with d = 0.0824742 (left plot) and d = 0.3 (right plot).

The power of the fundamental FEL harmonic is the same for x- and y-polarizations: ∼4 GW. Note in Fig. 10 that the powers of the high harmonics are rather weak, but for the fifth harmonic the power reaches 0.2 MW for d = 0.3 and 0.5 kW for d = 0.08247. The second harmonic has a power of ∼45 kW. It is evident upon comparison of the plots in Fig. 10 that the third and the fifth FEL harmonics are stronger in the undulator (18) for higher values of d, i.e. for stronger undulator field harmonics. The bunching evolution, corresponding to the FEL power evolution in the right-hand plot in Fig. 10, is demonstrated in Fig. 11.

Figure 11
Bunching evolution in the cos–sin spiral undulator FEL with the field (18)

for γ = 1570, I₀ = 120 A, ∊ⁿ = 10⁻⁶ mm mrad, β = 0.37 m, k = 2.216, h = 3, d = 0.3, λ_u = 2.3 cm. Harmonics: n = 1 (red), n = 2 (yellow), n = 3 (green), n = 4 (green dashed), n = 5 (blue dotted).

We also studied the radiation from the FEL with the bi-harmonic elliptic undulator (18) and SACLA installation beam. Assuming SACLA standard focusing, β = 5 m, the second harmonic is quite weak (see Fig. 12); the fifth FEL harmonic is weak, but it is amplified due to the third undulator field harmonic [see (18)], especially for d = 0.3 (see blue dotted lines in Figs. 12 and 10). The modeling data for the FEL with the SACLA beam and the undulator (18) for d = 0.08247 are as follows: γ = 1570, P_E = 96.27 GW, J = 6 × 10⁹ A, Σ_beam = 2 × 10⁻⁸ m², I₀ = 120 A, σ_e = 0.0002, ∊ⁿ = 10⁻⁶ µm rad, β = 5 m, ζ = 1.05, k = 2.21622, h = 3, d = 0.08247, λ_u = 2.3 cm, L_s = 11.8 m, L_g = 0.97 m, λ₁ = 27.6 nm, λ₂ = 13.8 nm, λ₃ = 9.2 nm, λ₄ = 6.90 nm, λ₅ = 5.52 nm,

$[\eqalign{ f_{n\semi y,x} & = \{0.9974,0.0657,0.0113,0.0056,0.0271\}, \cr \rho_{n\semi y,x} & \simeq \{0.00117,0.00016,0.00004,0.000025,0.00009\}, \cr P_{n,y,x,F} & \simeq \{1.37\times10^8, 28.7\times10^3, 0.8, 0, 220\}\,\,{\rm{W}}. } ]$

Figure 12
Harmonic power evolution in the cos–sin elliptic undulator FEL with the field (18)

for the SACLA beam: β = 5 m, k = 2.21622, h = 3, λ_u = 2.3 cm; both x- and y-polarization have the same power. Harmonics: n = 1 (red), n = 2 (yellow), n = 3 (green), n = 4 (green dashed), n = 5 (blue dotted). Undulator with d = 0.0824742 (left plot) and d = 0.3 (right plot).

The gain length of the considered bi-harmonic elliptic FEL is L_g ≃ 1 m and the saturation length is L_s ≃ 12 m independent of d. The saturated power of the fundamental tone reaches 0.14 GW. Already for d ≃ 0.08 the fifth UR harmonic gets some boost (see left-hand plot in Fig. 12); for d = 0.3 its power exceeds 10 kW and the third UR harmonic appears (see right-hand plot in Fig. 12). Thus, a relatively weak third undulator field harmonic with d ≃ 0.1–0.3 and off-axis effects can generate noticeable second and fifth UR harmonics of the spontaneous and stimulated radiation. The effect of the undulator field harmonic is stronger on the spontaneous UR than on the FEL radiation. The fifth UR harmonic is stronger than the third.

6. Conclusions

We analyzed the influence of the third harmonic of the undulator field on the spontaneous and stimulated radiation for planar and elliptic undulators. We obtained exact analytical expressions for the Bessel coefficients. They depend on the undulator parameter k and on the third field harmonic rate d in (1) and in (18). For the planar undulator we found that if k < 1.5 then the third field harmonic with d < 0 (negative phase) enhances UR harmonics with n = 3, 5,…, and for d > 0 (positive phase) the power of the high harmonics decreases (see Fig. 1). If k > 1.5, then the effect of the magnetic field harmonic in (1) is the opposite: high UR harmonics n = 3, 5,… are stronger for d > 0 and weaker for d < 0 (see Fig. 1). This observation corrects and clarifies previous reports in the literature (Zhukovsky, 2015a,b, 2016a,b; Jeevakhan & Mishra, 2011; Mishra et al., 2009; Jia, 2011).

The effect of the energy spread σ_e ≃ 10⁻⁴ on the UR is small; for σ_e ≃ 10⁻³, high UR harmonic power decreases as expected (see Fig. 2). All of the above results were confirmed by numerical simulations using the SPECTRA program: see Figs. 3 and 4.

We further developed the phenomenological FEL model, including the description of the harmonic behavior around the saturation region and the multi-stage saturation, which matches the experiments. We studied the harmonic generation in several setups of SPring-8, SACLA and LCLS FEL installations. Our modeling (see Fig. 7) agreed with the experimental measurements, with our numerical simulations in PERSEO (Zhukovsky & Kalitenko, 2019c ,d ) and with the results using the GENESIS program. The match for the third and fifth harmonic powers was remarkably good. The second harmonic power had fairly good agreement with the experiments too.

We explored the influence of the third undulator field harmonic on the FEL radiation. For a planar undulator with k ≃ 2 the effect is small for any d; for k ≃ 3.5 it is noticeable for d = ±0.3. For k ≃ 1 and, in particular, for k < 1, the effect of the second field in (1) can be significant. We studied the SPring-8 FEL with γ = 489, σ_e = 2 × 10⁻⁴, ∊_x,yⁿ = 2.7 mm mrad, k = 0.7, λ_u = 1.5 cm, I = 300 A, fundamental wavelength λ₁ = 40 nm (Shintake et al., 2009); the change of the power of the third harmonic is ∼200% for d = ±0.3 (see Fig. 8). The change of the harmonic power for n = 5 can be ∼10² times (see Fig. 8). The second FEL harmonic is influenced mostly by the finite beam size and off-axis effect. The newly updated SACLA setup is not sensitive to the third undulator field harmonic.

We studied the third field harmonic effect in an elliptic undulator (18). Even a relatively weak third undulator field harmonic, d ≃ 0.08, in (18) gives rise to a noticeable fifth UR harmonic (see Figs. 9 and 10). The fifth UR harmonic is stronger than the third. Their powers further increase by two orders of magnitude if d = 0.3. Due to the finite electron beam size and divergence, the radiation of the second harmonic can be significant; it can prevail over the fifth harmonic, whose content is ∼0.01% (see Fig. 12).

The above analysis may help to identify the degree of harmonic presence in FEL radiation, attributed to the undulator field harmonics.

Acknowledgements

We are grateful to Professor A. V.Borisov for useful notes and to A. M. Kalitenko for the SPECTRA data and his help in checking some of the Bessel functions.

References

Alesini, D., Bertolucci, S., Biagini, M. E., Biscari, C., Boni, R., Boscolo, M., Castellano, M., Clozza, A., Di Pirro, G., Drago, A., Esposito, A., Ferrario, M., Fusco, V., Gallo, A., Ghigo, A., Guiducci, S., Incurvati, M., Ligi, C., Marcellini, F., Migliorati, M., Milardi, C., Mostacci, A., Palumbo, L., Pellegrino, L., Preger, M., Raimondi, P., Ricci, R., Sanelli, C., Serio, M., Sgamma, F., Spataro, B., Stecchi, A., Stella, A., Tazzioli, F., Vaccarezza, C., Vescovi, M., Vicario, C., Zobov, M., Alessandria, F., Bacci, A., Boscolo, I., Broggi, F., Cialdi, S., DeMartinis, C., Giove, D., Maroli, C., Mauri, M., Petrillo, V., Romè, M., Serafini, L., Levi, D., Mattioli, M., Medici, G., Catani, L., Chiadroni, E., Tazzari, S., Bartolini, R., Ciocci, F., Dattoli, G., Doria, A., Flora, F., Gallerano, G. P., Giannessi, L., Giovenale, E., Messina, G., Mezi, L., Ottaviani, P. L., Pagnutti, S., Picardi, L., Quattromini, M., Renieri, A., Ronsivalle, C., Cianchi, A., Angelo, A. D., Di Salvo, R., Fantini, A., Moricciani, D., Schaerf, C. & Rosenzweig, J. B. (2004). Nucl. Instrum. Methods Phys. Res. A, 528, 586–590. Web of Science CrossRef CAS Google Scholar
Dattoli, G., Giannessi, L., Ottaviani, P. L. & Ronsivalle, C. (2004). J. Appl. Phys. 95, 3206–3210. Web of Science CrossRef CAS Google Scholar
Dattoli, G. & Ottaviani, P. L. (2002). Opt. Commun. 204, 283–297. Web of Science CrossRef CAS Google Scholar
Dattoli, G., Ottaviani, P. L. & Pagnutti, S. (2005b). J. Appl. Phys. 97, 113102. Web of Science CrossRef Google Scholar
Dattoli, G., Ottaviani, P. L. & Renieri, A. (2005a). Laser Part. Beams, 23, 303. Web of Science CrossRef Google Scholar
Emma, P. et al. (2010). Nat. Photon. 4, 641–647. Web of Science CrossRef CAS Google Scholar
Giannessi, L. et al. (2011). Phys. Rev. ST Accel. Beams, 14, 060712. Web of Science CrossRef Google Scholar
Huang, Z. & Kim, K.-J. (2007). Phys. Rev. ST Accel. Beams, 10, 034801. Web of Science CrossRef Google Scholar
Jeevakhan, H. & Mishra, G. (2011). Nucl. Instrum. Methods Phys. Res. A, 656, 101–106. Web of Science CrossRef CAS Google Scholar
Jia, Q. (2011). Phys. Rev. ST Accel. Beams, 14, 060702. Web of Science CrossRef Google Scholar
Lee, K., Mun, J., Hee Park, S., Jang, K., Uk Jeong, Y. & Vinokurov, N. A. (2015). Nucl. Instrum. Methods Phys. Res. A, 776, 27–33. Web of Science CrossRef CAS Google Scholar
McNeil, B. W. J. & Thompson, N. R. (2010). Nat. Photon. 4, 814–821. Web of Science CrossRef CAS Google Scholar
Margaritondo, G. & Rebernik Ribic, P. (2011). J. Synchrotron Rad. 18, 101–108. Web of Science CrossRef CAS IUCr Journals Google Scholar
Milton, S. V., Gluskin, E., Arnold, N. D., Benson, C., Berg, W., Biedron, S. G., Borland, M., Chae, Y. C., Dejus, R. J., Den Hartog, P. K., Deriy, B., Erdmann, M., Eidelman, Y. I., Hahne, M. W., Huang, Z., Kim, K. J., Lewellen, J. W., Li, Y., Lumpkin, A. H., Makarov, O., Moog, E. R., Nassiri, A., Sajaev, V., Soliday, R., Tieman, B. J., Trakhtenberg, E. M., Travish, G., Vasserman, I. B., Vinokurov, N. A., Wang, X. J., Wiemerslage, G. & Yang, B. X. (2001). Science, 292, 2037–2041. Web of Science CrossRef PubMed Google Scholar
Mishra, G., Gehlot, M. & Hussain, J. K. (2009). Nucl. Instrum. Methods Phys. Res. A, 603, 495–503. Web of Science CrossRef CAS Google Scholar
Owada, S., Togawa, K., Inagaki, T., Hara, T., Tanaka, T., Joti, Y., Koyama, T., Nakajima, K., Ohashi, H., Senba, Y., Togashi, T., Tono, K., Yamaga, M., Yumoto, H., Yabashi, M., Tanaka, H. & Ishikawa, T. (2018). J. Synchrotron Rad. 25, 282–288. Web of Science CrossRef CAS IUCr Journals Google Scholar
Pellegrini, C. (2016). Phys. Scr. T169, 014004. Web of Science CrossRef Google Scholar
Pellegrini, C., Marinelli, A. & Reiche, S. (2016). Rev. Mod. Phys. 88, 015006. Web of Science CrossRef Google Scholar
Ratner, D., Brachmann, A., Decker, F. J., Ding, Y., Dowell, D., Emma, P., Fisher, A., Frisch, J., Gilevich, S., Huang, Z., Hering, P., Iverson, R., Krzywinski, J., Loos, H., Messerschmidt, M., Nuhn, H. D., Smith, T., Turner, J., Welch, J., White, W. & Wu, J. (2011). Phys. Rev. ST Accel. Beams, 14, 060701. Web of Science CrossRef Google Scholar
Saldin, E. L., Schneidmiller, E. A. & Yurkov, M. V. (2000). The Physics of Free Electron Lasers. Springer. Google Scholar
Shintake, T. et al. (2009). Phys. Rev. ST Accel. Beams, 12, 070701. Web of Science CrossRef Google Scholar
Tanaka, T. (2014). Phys. Rev. ST Accel. Beams, 17, 060702. Web of Science CrossRef Google Scholar
Tanaka, T. & Kitamura, H. (2001). J. Synchrotron Rad. 8, 1221–1228. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zhukovsky, K. V. (2015a). Moscow Univ. Phys. 70, 232–239. Web of Science CrossRef Google Scholar
Zhukovsky, K. (2015b). Opt. Commun. 353, 35–41. Web of Science CrossRef CAS Google Scholar
Zhukovsky, K. (2016a). Laser Part. Beams 34, 447. Web of Science CrossRef Google Scholar
Zhukovsky, K. (2016b). Nucl. Instrum. Methods Phys. Res. B, 369, 9–14. Web of Science CrossRef CAS Google Scholar
Zhukovsky, K. (2017a). J. Phys. D Appl. Phys. 50, 505601. Web of Science CrossRef Google Scholar
Zhukovsky, K. J. (2017b). J. Appl. Phys. 122, 233103. Web of Science CrossRef Google Scholar
Zhukovsky, K. V. (2018). Russ. Phys. J. 60, 1630–1637. Web of Science CrossRef CAS Google Scholar
Zhukovsky, K. V. (2019). Results Phys. 13, 102248. Web of Science CrossRef Google Scholar
Zhukovsky, K. & Kalitenko, A. (2019a). J. Synchrotron Rad. 26, 159–169. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zhukovsky, K. & Kalitenko, A. (2019b). J. Synchrotron Rad. 26, 605–606. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zhukovsky, K. & Kalitenko, A. M. (2019c). Radiophys. Quantum Electron. 62, 52–64. Google Scholar
Zhukovsky, K. & Kalitenko, A. M. (2019d). Russ. Phys. J. 62, 354–362. Web of Science CrossRef Google Scholar
Zhukovsky, K. & Potapov, I. (2017). Laser Part. Beams 35, 326. Web of Science CrossRef Google Scholar

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JOURNAL OF
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RADIATION

ISSN: 1600-5775

Volume 26| Part 5| September 2019| Pages 1481-1488

https://doi.org/10.1107/S1600577519008415

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Effect of the third undulator field harmonic on spontaneous and stimulated undulator radiation

1. Introduction

2. Spontaneous UR and validation of the Bessel factors

3. Phenomenological description of FEL harmonics evolution

4. Effect of the third field harmonic on the planar FEL radiation

5. Effect of the third field harmonic on the elliptic FEL radiation

6. Conclusions

Acknowledgements

References

research papers