Electron beam energy spread measurements using optical klystron radiation

In accelerators, the electron beam longitudinal dynamics critically depend on the energy distribution of the beam. Noninvasive, highly accurate measurement of the energy spread of the electron beam in the storage ring remains a challenge. Conventional techniques are limited to measuring a relatively large energy spread using the energy spread induced broadening effect of radiation source size or radiation spectrum. In this work, we report a versatile method to accurately measure the electron beam relative energy spread from ${10}^{-4}$ to ${10}^{-2}$ using the optical klystron radiation. A novel numerical method based on the Gauss-Hermite expansion has been developed to treat both spectral broadening and modulation on an equal footing. A large dynamic range of the measurement is realized by properly configuring the optical klystron. In addition, a model-based scheme has been developed for the first time to compensate the beam-emittance-induced inhomogeneous spectral broadening effect to improve the accuracy of the energy spread measurement. Using this technique, we have successfully measured the relative energy spread of the electron beam in the Duke storage ring from $6\times {10}^{-4}$ to $6\times {10}^{-3}$ with an overall uncertainty of less than 5%. The optical klystron is a powerful diagnostic for highly accurate energy spread measurement for storage rings and other advanced electron accelerators.

Figure 1

Simulated optical klystron spectrum and fitted curves with both G-H and SES methods. The simulated spectrum is generated with $N_d=31$, ${\sigma }_{\mathrm{E}}^{\mathrm{set}}=5.00\times {10}^{-4}$, and ${ϵ}_x=1\mathrm{nm}$; the other beam parameters are listed in Table . The fitted curve using the G-H method and the fitted curve using the SES method are indistinguishable and both agree well with the simulated spectrum. A single undulator spectrum ($N_u=33$) with a monoenergetic beam is also shown as an expected envelope of the optical klystron spectrum when the energy spread is small.

Figure 2

A simulated spectrum and related fitted curves. The simulated spectrum is generated with ${\sigma }_{\mathrm{E}}^{\mathrm{set}}=2.80\times {10}^{-3}$, ${ϵ}_x=1\mathrm{nm}$, and $N_d=31$. The red solid curve is a fitted spectrum using the G-H method; the black dashed curve is the fitted spectrum using the SES method. The fitted values of the energy spread using both methods are listed in the plot.

Figure 3

Comparison of the fitted energy spread ${\sigma }_{\mathrm{E}}^{\mathrm{fit}}$ and set energy spread ${\sigma }_{\mathrm{E}}^{\mathrm{set}}$. Parameters to generate simulated spectra are listed in Table  with $N_d=31$ and ${ϵ}_x=1\mathrm{nm}$. The fitted values of the energy spread using the G-H method are plotted as circles, while the fitted values using the SES method are plotted as crosses. The error bars show the RMS uncertainty of the fitting. The inset shows the relative difference between the fitted and set values of the energy spread, $\Delta {\sigma }_E^{\mathrm{diff}}$ [Eq. (11)].

Figure 4

A simulated spectrum and related fitted curves. Parameters to generate simulated spectra are listed in Table  with ${\sigma }_{\mathrm{E}}^{\mathrm{set}}=5.00\times {10}^{-3}$, $N_d=1$, and ${ϵ}_x=1\mathrm{nm}$. The red solid curve is a fitted spectrum using the G-H method; the black dashed curve is the fitted spectrum using the SES method. The fitted values of the energy spread using both methods are listed in the plot.

Figure 5

Comparison of the fitted and set values of the energy spread in three regions: R1 (${\sigma }_{\mathrm{E}}^{\mathrm{set}}<{\sigma }_L$), R2 (${\sigma }_L<{\sigma }_{\mathrm{E}}^{\mathrm{set}}<{\sigma }_H$), and R3 (${\sigma }_{\mathrm{E}}^{\mathrm{set}}>{\sigma }_H$). Parameters to generate simulated spectra are listed in Table  with ${ϵ}_x=1\mathrm{nm}$. For small energy spread, $N_d=31$, and for large energy spread, $N_d=1$. The G-H method is used to fit the spectra in both R1 and R2 regions; the simple Gaussian fit is used for some spectra in the region R2 and all the spectra in the region R3. The inset shows the relative discrepancy between the fitted and set values of the energy spread in the three regions.

Figure 6

Illustration of the electron trajectory and emission in an optical klystron with planar undulators. The electron is moving in the midplane of the undulator field with a horizontal angle $p_x$ (the normalized horizontal momentum) with respect to the longitudinal $z$ axis. The radiation is collected behind an aperture opening with half-angle ${\theta }_{ap}$. The radiation is collected at a particular opening angle $\theta$ with respect to the electron trajectory.

Figure 7

Comparison of the fitted and set values of the energy spread with large emittance values, ${ϵ}_x=40,45,50\mathrm{nm}$. Other simulation parameters are found in Table  with $N_d=31$. The inset shows the relative difference between the fitted and set values, ${\sigma }_{\mathrm{E}}^{\mathrm{diff}}$ [Eq. (11)].

Figure 8

Constrained parameter space of (${\sigma }_{\mathrm{E}}^{\mathrm{set}},{ϵ}_x$), the shaded area, is used to determine the emittance correction formula. The shaded area is defined by $a_x\in [1,4.5]$ and $4.00\times {10}^{-4}\le {\sigma }_{\mathrm{E}}^{\mathrm{set}}\le 1.60\times {10}^{-3}$.

Figure 9

Additional spectrum broadening, ${\sigma }_{\mathrm{E},{ϵ}_x}$, as a function of the horizontal electron beam emittance ${ϵ}_x$. ${\sigma }_{\mathrm{E},{ϵ}_x}$ is fit to a linear function of ${ϵ}_x$ and the fitted parameters are listed in the plot.

Figure 10

Comparison of the emittance-effect corrected energy spread ${\sigma }_{\mathrm{E}}^{\mathrm{corr}}$ and set value ${\sigma }_{\mathrm{E}}^{\mathrm{set}}$ for ${ϵ}_x=40,45,50\mathrm{nm}$. The inset shows the relative difference between ${\sigma }_{\mathrm{E}}^{\mathrm{corr}}$ and ${\sigma }_{\mathrm{E}}^{\mathrm{set}}$.

Figure 11

A schematic of the experimental setup to measure the electron beam energy spread using a diagnostic optical klystron system on the Duke storage ring.

Figure 12

The measured grating efficiency curve for the Ocean Optics HR4000 spectrometer. The efficiency curve is calibrated using known spontaneous undulator spectra which are varied to cover the wavelength range of the spectrometer. The efficiency curve converges after a few iterations. The dotted curve is the efficiency curve after the first iteration; and the solid curve is the converged efficiency curve after two iterations.

Figure 13

A typical measured optical klystron spectrum before and after the correction using the grating efficiency curve. The raw spectrum is represented by blue crosses while the corrected spectrum is shown using red circles.

Figure 14

Measured optical klystron spectra with a varying energy spread and a high buncher setting $N_d=31$. The electron beam energy is 400 MeV and the single-bunch beam current is about 15 mA. The FEL interaction is adjusted to increase the energy spread of the beam. Both the G-H method and the SES method are used to fit the measured spectra. The fitted values of the energy spread using both methods are listed in the plot. In each subplot, the background of the measured spectrum, $h$ in Eq. (6), has been subtracted.

Figure 15

The transverse beam image measured at the synchrotron radiation port. The electron beam energy is 400 MeV and single-bunch beam current is 15 mA. Horizontal and vertical projections of the beam transverse profile are also shown. A Gaussian fit of the horizontal beam distribution yields ${\sigma }_x=0.16\mathrm{mm}$.

Figure 16

Measured optical klystron spectra with an increased energy spread and a low buncher setting $N_d=1.09$. The electron beam energy is 400 MeV and the single-bunch beam current is about 15 mA. Compared to Fig. 14, the FEL interaction is increased to produce a larger energy spread. Both the G-H method and the SES method are used to fit the measured spectra. The fitted values of the energy spread using both methods are listed in the plot. In each subplot, the background of the measured spectrum has been subtracted.

Figure 17

Measured optical klystron spectrum with a very small energy spread. The electron beam energy is 280 MeV and the single-bunch beam current is 0.045 mA. To produce substantial spectral intensity modulation, the buncher magnet is powered with a large current with $N_d\approx 56$. The G-H method is used to find the energy spread of the beam, ${\sigma }_{\mathrm{E}}^{\mathrm{fit}}=(6.00\pm 0.015)\times {10}^{-4}$. The background of the measured spectrum has been subtracted.

Figure 18

Measured optical klystron spectrum with a very large energy spread. The electron beam energy is 400 MeV and the single-bunch beam current is 40 mA. The FEL interaction is turned on to increase the energy spread. The buncher magnet is powered with a small current with $N_d=1.06$. The G-H method is used to find the energy spread of the beam, ${\sigma }_{\mathrm{E}}^{\mathrm{fit}}=(6.00\pm 0.029)\times {10}^{-3}$. The background of the measured spectrum has been subtracted.