4 FEL Physics. Technical Synopsis

Transcription

1 4 FEL Physics Technical Synopsis This chapter presents an introduction to the Free Electron Laser (FEL) physics and the general requirements on the electron beam parameters in order to support FEL lasing and saturation for output photon energies in the 250 ev to 13-keV range. It also describes the FEL transverse and temporal properties in the Self Amplified Spontaneous Emission (SASE) mode, which is the baseline for the LCLS-II. April 8, 2011 SLAC-I R

2 4.1 Introduction and Overview An FEL is a tunable source of coherent radiation that uses a bunch of relativistic electrons to resonantly amplify an electromagnetic wave through an undulator. The initial electromagnetic wave can be an external source that co-propagates with the electron bunch or the spontaneous emission generated by random electron microbunching (i.e., shot noise) in the undulator. Resonant interaction between the wave, the undulator field, and the electron bunch creates coherent electron density modulations on the scale of the resonant wavelength, which then lead to the exponential growth of the radiation power at this wavelength in a long undulator. The exponential growth in radiation power ceases when the FELinduced energy spread on the electron bunch stops and then begins to decrease the FEL microbunching. The first lasing and operation of the LCLS-I [1] has produced results in very good agreement with standard FEL theory and numerical simulations, indicating that FEL physics in the x-ray wavelength range is well understood. The LCLS-II FEL design is closely based on the very successful LCLS-I FEL design. The main differences are the use of variable-gap undulators instead of fixed-gap undulators, as well as the arrangement of two undulator lines in parallel. The baseline LCLS-II will operate in SASE mode for both the hard and soft x-ray FELs. Future expansion options for the LCLS-II include polarization control and seeding for both FELs. The basic electron beam requirements are discussed in Section 4.2 and the expected x-ray FEL characteristics are discussed in Section Electron Beam Requirements A relativistic electron beam entering the undulator will generate spontaneous undulator radiation. If the beam energy is mc 2, the undulator period is u, and the undulator parameter is K, the wavelength of the undulator radiation in the forward direction is The undulator parameter K is defined as 2 u K (1) eb K 0 u B0[ Tesla ] [ ] 2 mc cm u, (2) where B 0 is the peak magnetic field for a planar undulator. For a variable-gap undulator (see Chapter 7), the magnetic field can be changed by adjusting the undulator gap, and the typical field strength can be varied from a few kg to slightly above 1 Tesla. For a permanent magnet undulator that has a few cmlong period, the undulator parameter K varies between 1 and 10. For the LCLS-II hard x-ray FEL, undulator parameters have been chosen such that a 13 GeV beam is necessary to generate 1 Angstrom wavelength radiation. The spontaneous radiation can be amplified by the FEL instability if the electron beam quality is sufficiently high; this process has been called self-amplified spontaneous emission (SASE) [2,3]. High gain FELs are often characterized by the FEL Pierce parameter [3] 1 16 I I pk A K [ JJ] u x 1/ 3 1 I 4 I pk A K [ JJ] (1 K / 2) 4 x 1/ 3. (3) April 8, 2011 SLAC-I R

3 Here I pk is the electron peak current, I A =17 ka, [JJ] is the Bessel function coupling factor associated with a planar undulator, is the electron beam Lorentz factor, and x is the rms electron beam size in the undulator. The SASE process is largely determined by the FEL Pierce parameter, which ranges from for soft x-ray FELs to for hard x-ray FELs. For example, the power gain length is L G ( 1) LG 0 with u LG 0, (4) 4 3 where is a gain degradation parameter due to radiation diffraction, beam emittance and energy spread [4]. The FEL saturation power is roughly P beam, where P beam is the electron beam power. Beginning from shot noise, it typically takes a SASE FEL about 18 to 20 L G to reach saturation. The electron beam quality has to be extremely high in order to obtain a relatively short FEL gain length (i.e., in Equation (4) should be smaller than unity). For a typical x-ray FEL, this requires I e ~ a few ka, x,y ~ /(4), <<. (5) Here all beam parameters refer to their slice values (i.e., those corresponding to temporal slice whose length is that of a typical SASE spike as described in Section 4.3. The high peak current electron bunch can be obtained by magnetic bunch compression upstream in the linac. The unnormalized emittance in Equation (5) is determined by the final beam energy, the injector performance, and control of any emittance degradation that can occur in the bunch compressors and other dipole optics, or might arise from wakefields associated with the linac cavities. The relative energy spread is determined by the final beam energy, intentional growth via a laser heater (to control the longitudinal microbunching instability) just downstream from the injector, and by conservation of longitudinal emittance in the bunch compression process. The FEL design parameters and simulation studies will be given in Chapter 5. The chosen accelerator design to achieve the required electron beam beam brightness will be presented in Chapter FEL X-Ray Properties The SASE FEL radiation from a planar undulator is linearly polarized in the plane of the electron s wiggle motion. The resonant interaction between the relativistic electrons, the undulator and radiation will amplify spontaneous emission that lies near the fundamental undulator radiation frequency. The typical SASE bandwidth at FEL saturation is [5] 3 4. (6) ~ 310 to510 Although many transverse modes can be excited at the beginning of the undulator, by the end of the exponential growth regime a fundamental mode with the highest growth rate (generally the TEM00 mode) will dominate. As a result, a SASE FEL has excellent transverse coherence at saturation and can usually be approximated by a fundamental Gaussian mode. In the x-ray wavelength range, the rms radiation mode size in the exponential growth regime can be estimated by [6] r x D, with D LG0. (7) 4 April 8, 2011 SLAC-I R

4 The rms divergence is then /(4 ) r' r /(4 L 1/ 4 G0 3/ 4 ) 1/ 2 x. (8) After FEL saturation, the radiation divergence stays more or less constant, while the rms mode size starts to increase due to diffraction. Therefore, Equation (8) can be regarded as a lower limit of the FEL divergence angle. Due to its startup from shot noise, the temporal behavior and fluctuations of a SASE FEL are those of chaotic, polarized light. The SASE light consists of random temporal spikes whose characteristic duration is the coherence time [7]. Close to FEL saturation, the coherence time is given by c 2 / c. (9) If the radiation pulse duration T is much longer than c, the radiation will consist of M > 1 independent modes. The shot-to-shot statistical fluctuation of the SASE pulse energy W in the exponential growth regime is then given by W W 1 M, where T M. (10) In the frequency domain, the SASE spectrum also exhibits spiky behavior, with M independent modes falling within the full normalized spectral bandwidth ~2. For the nominal LCLS-II 250 pc bunch charge and 80 fs bunch duration, M can be on the order 100. The statistical fluctuation before saturation can thus be on the level of 10%. Saturation effects reduce the statistical fluctuation to a much smaller level. In addition to statistical fluctuations, the output SASE pulse energy will fluctuate when the machine parameters change from shot to shot. For baseline design parameters with M~100, we expect that that shot-to-shot machine variations typically will be the dominant source of fluctuations. In a high-gain FEL near saturation, strong microbunching at the fundamental wavelength can drive substantial levels of harmonic microbunching. For a planar undulator, this harmonic microbunching can lead to significant radiation at the odd harmonics being generated in the forward direction, with the third harmonic power reaching the 1%-level relative to the fundamental power at saturation [6]. The relative spectral bandwidth of the harmonic radiation is similar to that of the fundamental. Since even harmonics can only be emitted off-axis for aligned electron beams, the coherent radiation from even harmonic microbunching is largely suppressed. For example, the 2 nd harmonic content is expected to be on the order of 10-4 as compared to the fundamental power level. The baseline LCLS-II project will construct two SASE-based undulator lines. To reduce the output radiation bandwidth and to suppress the spiky temporal structure arising from the SASE process, possible future LCLS-II upgrades include self-seeding and external seeding options. Polarization control units could be added after the main planar undulator to provide flexible x-ray polarization. These future upgrade options are discussed in Appendix D. c April 8, 2011 SLAC-I R

5 4.4 References 1. P. Emma et al., First lasing and operation of an ångstrom-wavelength free-electron laser, Nature Photonics 4, 641 (2010). 2. K. Kondratenko and E. Saldin, Generating of coherent radiation by a relativistic electron beam in an ondulator, Part. Accel. 10, 207 (1980). 3. R. Bonifacio, C. Pellegrini, and L. Narducci, Collective instabilities and high-gain regime in a free electron laser, Opt. Commun. 50, 373 (1984). 4. L.-H. Yu, S. Krinsky, R. Gluckstern, Calculation of universal scaling function for free-electron-laser gain, Phys. Rev. Lett. 64, 3011 (1990); M. Xie, Design Optimization for an x-ray free electron laser driven by SLAC linac, in Proceedings of the 1995 Particle Accelerator Conference (IEEE, Piscataway, NJ, 1995), p K.-J. Kim, Three-dimensional analysis of coherent amplification and self-amplified spontaneous emission in free-electron lasers, Phys. Rev. Lett. 57, 1871 (1986); J.-M. Wang and L.-H. Yu, A transient analysis of a bunched beam free electron laser, Nucl. Instrum. Methods A 250, 484 (1986). 6. Z. Huang and K.-J. Kim, Review of x-ray free-electron laser theory, Phys. Rev. ST Accel. Beams 10, (2007). 7. E. Saldin, E. Schneidmiller, and M. Yurkov, Statistical properties of radiation from VUV and X-ray free electron laser, Opt. Commun. 148, 383 (1998). April 8, 2011 SLAC-I R