Harmonic lasing in x-ray free electron lasers

Harmonic lasing in a free electron laser with a planar undulator (under the condition that the fundamental frequency is suppressed) might be a cheap and efficient way of extension of wavelength ranges of existing and planned x-ray free electron laser (FEL) facilities. Contrary to nonlinear harmonic generation, harmonic lasing can provide much more intense, stable, and narrow-band FEL beam which is easier to handle due to the suppressed fundamental frequency. In this paper we perform a parametrization of the solution of the eigenvalue equation for lasing at odd harmonics, and present an explicit expression for FEL gain length, taking into account all essential effects. We propose and discuss methods for suppression of the fundamental harmonic. We also suggest a combined use of harmonic lasing and lasing at the retuned fundamental wavelength in order to reduce bandwidth and to increase brilliance of x-ray beam at saturation. Considering 3rd harmonic lasing as a practical example, we come to the conclusion that it is much more robust than usually thought, and can be widely used in the existing or planned x-ray FEL (XFEL) facilities. In particular, Linac Coherent Light Source (LCLS) after a minor modification can lase to saturation at the 3rd harmonic up to the photon energy of 25–30 keV providing multigigawatt power level and narrow bandwidth. As for the European XFEL, harmonic lasing would allow one to extend operating range (ultimately up to 100 keV), to reduce FEL bandwidth and to increase brilliance, to enable two-color operation for pump-probe experiments, and to provide more flexible operation at different electron energies. Similar improvements can be realized in other x-ray FEL facilities with gap-tunable undulators like FLASH II, SACLA, LCLS II, etc. Harmonic lasing can be an attractive option for compact x-ray FELs (driven by electron beams with a relatively low energy), allowing the use of the standard undulator technology instead of small-gap in-vacuum devices. Finally, in this paper we discover that in a part of the parameter space, corresponding to the operating range of soft x-ray beam lines of x-ray FEL facilities (like SASE3 beam line of the European XFEL), harmonics can grow faster than the fundamental wavelength. This feature can be used in some experiments, but might also be an unwanted phenomenon, and we discuss possible measures to diminish it.

Figure 1

Coupling factors for the 1st, 3rd, and 5th harmonics (denoted with 1, 3, and 5, correspondingly) versus rms undulator parameter.

Figure 2

Ratio of gain lengths of the retuned fundamental and the third harmonic for lasing at the same wavelength versus rms undulator parameter $K$. The fundamental wavelength is reduced by means of reducing the undulator parameter $K$ (solid) or increasing beam energy (dash).

Figure 3

Parameter $\delta$ for the retuned fundamental harmonic (${\delta }^{(1)}$, solid line), and for the third harmonic (${\delta }^{(3)}$, dashed line), versus rms undulator parameter. The gain lengths of the third harmonic and of the retuned fundamental harmonic are equal. Retuning is done by reducing undulator parameter.

Figure 4

Ratio of gain lengths for lasing at the fundamental wavelength and at the third harmonic versus diffraction parameter of the fundamental wavelength for large values of the undulator parameter $K$.

Figure 5

Diffraction parameter of the fundamental wavelength, for which the third (solid) and the fifth (dashed) harmonics have the same gain length as the fundamental, versus the rms undulator parameter $K$. Below these curves harmonics have shorter gain lengths than the fundamental frequency.

Figure 6

An example for the European XFEL. Averaged peak power for the fundamental harmonic (solid), the third harmonic (dashed), and the fifth harmonic (dotted) versus undulator length for SASE3 operating at 4.5 nm. Parameters are in the text. Simulations were performed with the code FAST.

Figure 7

Averaged peak power for the fundamental harmonic (solid) and the third harmonic (dash) versus geometrical length of the LCLS undulator (including breaks). The wavelength of the third harmonic is 0.5 Å (photon energy 25 keV). Beam and undulator parameters are in the text. The fundamental is disrupted with the help of the spectral filter (see the text) and of the phase shifters. The phase shifts are $4\pi /3$ after segments 1–5 and 17–22, and $2\pi /3$ after segments 6–10 and 23–28. Simulations were performed with the code FAST.

Figure 8

An example for the European XFEL. Averaged peak power for the fundamental harmonic (solid) and the third harmonic (dash) versus magnetic length of SASE1 undulator. The wavelength of the third harmonic is 0.2 Å (photon energy 62 keV). The fundamental is disrupted with the help of phase shifters installed after 5 m long undulator segments. The phase shifts are $4\pi /3$ after segments 1–8 and 21–26, and $2\pi /3$ after segments 9–16. Simulations were performed with the code FAST.

Figure 9

Normalized power of the fundamental harmonic (solid) and of the third harmonic (dash) versus normalized undulator length for a SASE FEL. Phase shifts are equal to $2\pi /3$ at the positions $\widehat{z}=4,5,6,\dots ,14$, as suggested in 18. The undulator parameter is large, $K\gg 1$. Definitions of the normalized parameters can be found in 44.

Figure 10

Normalized power of the fundamental harmonic (solid) and of the third harmonic (dash) versus normalized undulator length. Phase shifts are equal to $4\pi /3$ at the positions $\widehat{z}=1,2,3,8,9$ and $2\pi /3$ at the positions $\widehat{z}=4,5,6,7,10,11$. The undulator parameter is large, $K\gg 1$.

Figure 11

Ratio of gain lengths for lasing at the fundamental wavelength and at the third harmonic. The solid curve is calculated with the help of Eq. (e3) in the frame of the 3D model. The dashed curve is calculated with the help of Eq. (e1) in the frame of the 1D model. Adjustment of the fundamental wavelength was done by retuning of the undulator parameter $K$ according to (e2).