Revision of optical klystron enhancement effects in self-amplified spontaneous emission free electron lasers

We review the theory of optical klystrons, with its applications for self-amplified spontaneous emission free electron lasers in mind. We show that previous theories miss terms in the power gain factor that cannot be neglected, and we illustrate differences between the previously known analytical expressions, new ones found in this paper, and numerical calculations. We then consider the use of optical klystrons for electron energy-spread and radiation coherence-time diagnostics purposes.

Figure 1

Sketch of an optical klystron arrangement at a segmented FEL with tunable undulator gaps.

Figure 2

Numerical solutions (blue lines) of the eigenvalue equation for different values of ${\widehat{\mathrm{\Lambda }}}_T$. The dashed line corresponds to the cold-beam case. Note that only the values for $\widehat{C}$ corresponding to $\mathrm{Re}({\lambda }_1)>0$ are mathematically acceptable.

Figure 3

${\sigma }_{\widehat{C}}=0.1$. Left column: comparison of $G_1+G_2$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. Right column: comparison of $G_1+G_2+G_3$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. These different methods are rendered in the following way. Blue lines: numerical integration; black lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (35) and (36)]; yellow dashed lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a second order expansion in $\widehat{C}$ [Eqs. (a3) and (a4)]; red dashed lines: second effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (a8) and (a9)]; green lines: results from Ref. [2].

Figure 4

${\sigma }_{\widehat{C}}=0.5$. Left column: comparison of $G_1+G_2$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. Right column: comparison of $G_1+G_2+G_3$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. These different methods are rendered in the following way. Blue lines: numerical integration; black lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (35) and (36)]; yellow dashed lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a second order expansion in $\widehat{C}$ [Eqs. (a3) and (a4)]; red dashed lines: second effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (a8) and (a9)]; green lines: results from Ref. [2].

Figure 5

Left plot: value of $D{\widehat{\mathrm{\Lambda }}}_T$ for the maxima of $G_1+G_2$ in Fig. 3, as a function of ${\widehat{\mathrm{\Lambda }}}_T$ for the case ${\sigma }_{\widehat{C}}=0.1$. Right plot: the same for the maxima of $G_1+G_2$ in Fig. 4, that is for ${\sigma }_{\widehat{C}}=0.5$. Blue lines: numerical integration; black lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (35) and (36)]; yellow dashed lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a second order expansion in $\widehat{C}$ [Eqs. (a3) and (a4)]; red dashed lines: second effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (a8) and (a9)]; green lines: results from Ref. [2]. Only the analytical expression in Eq. (a4) depends, slightly, on ${\sigma }_{\widehat{C}}$; the blue lines (numerical integration) vary as well. Differences are very slight, although they can be seen by inspection.

Figure 6

Left plot: values of $D_{1/2}{\widehat{\mathrm{\Lambda }}}_T$ for $G_1+G_2$ (see text for further explanations) in Fig. 3, as a function of ${\widehat{\mathrm{\Lambda }}}_T$ for the case ${\sigma }_{\widehat{C}}=0.1$. Right plot: the same for the values of $D_{1/2}{\widehat{\mathrm{\Lambda }}}_T$ for $G_1+G_2$ in Fig. 4, that is for ${\sigma }_{\widehat{C}}=0.5$. Blue markers: numerical integration; black markers: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (35)–(37)]; yellow markers: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a second order expansion in $\widehat{C}$ [Eqs. (a3)–(a5)]; red markers: second effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (a8)–(a10)]; green markers: results from Ref. [2]. Only the analytical expression in Eq. (a4) depends, slightly, on ${\sigma }_{\widehat{C}}$; the blue markers (numerical integration) vary as well. Differences are very slight.

Figure 7

Value of ${\overline{G}}_{\max }/\overline{G}(0)$ at ${\sigma }_{\widehat{C}}=0.5$ (solid line). The dotted line is obtained fitting Eq. (40) to the numerical calculations.

Figure 8

Value of $D{\widehat{\mathrm{\Lambda }}}_T$ for which one obtains the maximum value of $G_1+G_2$, plotted against ${\widehat{\mathrm{\Lambda }}}_T$ for ${\sigma }_C=0.5$ (solid line). The dotted line is obtained fitting Eq. (41) to the numerical calculations.

Figure 9

Value of $D_{1/2}{\widehat{\mathrm{\Lambda }}}_T>0$, for which $\overline{G}$ drops to half of the value it has at $D=0$ for ${\sigma }_C=0.5$ (solid line). The dotted line is obtained fitting Eq. (42) to the numerical calculations.

Figure 10

${\sigma }_{\widehat{C}}=0.1$. Left column: comparison of $G_1+G_2$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. Center column: comparison of $G_1+G_2+G_3$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods. Right column: comparison of $\tilde{G}(D{\widehat{\mathrm{\Lambda }}}_T)/\tilde{G}(0)$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. These different methods are rendered in the following way. Blue lines: numerical integration; black lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (35)–(37)]; yellow lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a second order expansion in $\widehat{C}$ [Eqs. (a3)–(a5)]; red dashed lines: second effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (a8)–(a10)]; green lines: results from Ref. [2].

Figure 11

${\sigma }_{\widehat{C}}=0.5$. Left column: comparison of $G_1+G_2$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. Center column: comparison of $G_1+G_2+G_3$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods. Right column: comparison of $\tilde{G}(D{\widehat{\mathrm{\Lambda }}}_T)/\tilde{G}(0)$ at different values of ${\widehat{\mathrm{\Lambda }}}_T=0.1$, 0.2, 0.3, 0.4 (rows one to four) using different methods discussed in this paper. These different methods are rendered in the following way. Blue lines: numerical integration; black lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (35)–(37)]; yellow dashed lines: zeroth effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a second order expansion in $\widehat{C}$ [Eqs. (a3)–(a5)]; red dashed lines: second effective order in ${\widehat{\mathrm{\Lambda }}}_T\ll 1$, using a zeroth order expansion in $\widehat{C}$ [Eqs. (a8)–(a10)]; green lines: results from Ref. [2].