Transverse gradient undulator in a storage ring x-ray free electron laser oscillator

Modern electron storage rings produce bright x rays via spontaneous synchrotron emission, which is useful for a variety of scientific applications. The x-ray free-electron laser oscillator (XFELO) has the potential to amplify this output, both in terms of peak power and photon coherence. However, even current fourth generation storage rings (4GSRs) lack the requisite electron beam brightness to drive the XFELO due to its electron energy spread. The transverse gradient undulator (TGU) can overcome this issue, thus providing a practical means to couple the 4GSR to the XFELO. In this study, we first examine the theoretical basis of the TGU interaction by deriving the 3D TGU gain formula in the low gain approximation. Then, we perform an optimization study of the gain formula in order to determine optimal beam and machine parameters. Finally, we construct a hypothetical storage ring TGU-XFELO based on near-optimal parameters and report on its projected performance using multistage numerical simulation. We also discuss potential implementation challenges associated with the ring-FEL coupling.

Figure 1

Cross section of a transverse gradient undulator (TGU) aligned vertically. The direction of electron motion is in $z$. The magnetic field gradient grows stronger along the positive $y$ axis due to the canted undulator poles (exaggerated for visibility). If the electron bunch (center oval) is dispersed appropriately so that higher energy electrons (red; top half) experience higher field strength while lower energy electrons (blue; bottom half) experience less, then FEL gain can be greatly improved even for large electron energy spreads.

Figure 2

Schematic of the TGU-XFELO driven by an electron synchrotron. The three constituent components are the electron synchrotron (green), the TGU mechanism (blue), and the optical cavity (orange). The TGU mechanism consists of the undulator itself, as well as dispersive sections denoted by $D$ and $D^{-1}$ above.

Figure 3

Gain $G$ vs TGU parameter $\mathrm{\Gamma }$ depicting results from numerical integration (black, dashed) and time-independent genesis simulation (red, solid). Error bars indicate 2 standard deviations of the shot-to-shot variation. Simulation results consistently overperform when compared to theory, likely due to the magnitude of gain $G\sim \mathcal{O}(1)$ surpassing the low-gain assumption of the analytical formula. Nevertheless, both models are consistent in the location of the gain optimum at $\mathrm{\Gamma }=13.3$.

Figure 4

Gain $G$ versus electron beam dispersion $D$ (red, solid) and TGU magnetic gradient $\alpha$ (blue, dashed) derived from numerical integration of the 3D gain formula. These parameters are derived from the TGU parameter $\mathrm{\Gamma }$ by Eqs. (6) and (7). Optimal gain is attained at $D_{\mathrm{opt}}=6.2\mathrm{cm}$ and ${\alpha }_{\mathrm{opt}}=0.045{\mathrm{mm}}^{-1}$ for the chosen machine parameters.

Figure 5

Contour plot of gain as a function of transverse emittances ${ϵ}_x$, ${ϵ}_y$. White dashed lines represent levels of constant natural emittance ${ϵ}_{x,0}\equiv {ϵ}_x+{ϵ}_y$. For fixed ${ϵ}_{x,0}$, reducing the emittance ratio $k_c\equiv {ϵ}_y/{ϵ}_x$ (hence reducing ${ϵ}_y$) results in significant gain improvement. Hence it would be advantageous for the TGU to minimize $k_c$ for given ${ϵ}_{x,0}$.

Figure 6

Contour plot of gain vs electron and photon beam parameters in $x$ (right) and $y$ (left). In the $x$ direction, peak gain is located along the ${\beta }_x=Z_{Rx}$ line (black, dashed) as predicted by traditional FEL theory. In the $y$ direction, the contours are much more asymmetric due to the dispersion introduced by the TGU. In both directions, there are generous margins for potentially tuning these beam parameters while still maintaining respectable gain.

Figure 7

Flowchart of simulation pipeline used in the study. Blue lines indicate the electron simulation pathway, while orange lines indicate the x-ray pathway. The dashed line indicates an optional step. The run manager serves as a wrapper program that provides functions for initialization, beam generation/manipulation, reporting, and cleanup, among other things. The main FEL simulation is handled by modified genesis version 2 (see main text for details of modification), while elegant handles the storage ring propagation. Broken lines between genesis and elegant indicate that the electron file is not preserved between iterations (see text for discussion). X-ray cavity simulation is handled by an in-house Fourier-based code named pyopt. The x-ray file is preserved throughout.

Figure 8

Plot of XFELO pulse power vs turn number from numerical simulation. Maximum power of 15 MW is achieved at turn 95. The inset shows the frequency spectrum at peak power. The spectrum has a full-width half-maximum (FWHM) of 2 meV corresponding to a relative bandwidth of $1.4\times {10}^{-7}$.

Figure 9

X-ray beam spot at peak power. The root-mean-square (rms) beam sizes are ${\sigma }_x=14\mathrm{\mu }\mathrm{m}$ and ${\sigma }_y=12\mathrm{\mu }\mathrm{m}$. The slight asymmetry, despite the symmetric cavity, is due to the Bragg crystal filtering acting only in the horizontal plane.

Figure 10

Plot depicts the degradation of electron emittance ${ϵ}_y$ due to FEL gain within a single oscillator pulse. Initially, the low emittance ${ϵ}_y$ (red) enables a large positive gain (blue), which in turn exponentially amplifies the x-ray power (dashed gray; vertical scale not shown). The increasing x-ray power degrades ${ϵ}_y$ due to the TGU-FEL interaction. Near saturation, emittance ${ϵ}_y$ has increased so much that gain falls below zero, leading to the exponential decay of the x-ray pulse.

Figure 11

Proposed bunch train structure for ring-FEL coupling. Each vertical line represents one bunch, with its height proportional to peak current $I_{\mathrm{pk}}$ and thus TGU gain by Eq. (13). Tall red bunches are the designated “XFELO bunches” with $I_{\mathrm{pk}}$ exceeding the lasing threshold set by single-turn cavity loss. The temporal gap between XFELO bunches $T_{\mathrm{bunch}}$ is equal to the x-ray round-trip time $T_{\text{cavity}}$. The non-XFELO bunches (black) lay below the lasing threshold and only experience spontaneous undulator radiation. Thus, they are able to avoid substantial emittance degradation from the XFELO lasing process.