The physics of x-ray free-electron lasers

X-ray free-electron lasers (x-ray FELs) give us for the first time the possibility to explore structures and dynamical processes of atomic and molecular systems at the angstrom-femtosecond space and time scales. They generate coherent photon pulses with time duration of a few to 100 fs, peak power of 10 to 100 GW, over a wavelength range extending from about 100 nm to less than 1 Å. Using these novel and unique capabilities new scientific results are being obtained in atomic and molecular sciences, in areas of physics, chemistry, and biology. This paper reviews the physical principles, the theoretical models, and the numerical codes on which x-ray FELs are based, starting from a single electron spontaneous undulator radiation to the FEL collective instability of a high density electron beam, strongly enhancing the electromagnetic radiation field intensity and its coherence properties. A short review is presented of the main experimental properties of x-ray FELs, and the results are discussed of the most recent research to improve their longitudinal coherence properties, increase the peak power, and generate multicolor spectra.

Figure 1

Schematic representation of a free-electron laser.

Figure 2

Madey’s amplifier experiment. The undulator is a bifilar superconducting coil. From [44].

Figure 3

Madey’s FEL oscillator configuration. From [48].

Figure 4

Average FEL output energy, in nJ, as a function of electron beam peak current, compared with Ginger simulations for the UCLA–Los Alamos–Kurchatov FEL. The strong nonlinear dependence is in good agreement with SASE FEL theory. From [83].

Figure 5

The electron moves on the trajectory $P{P}^{\prime }$, the observer is at point $D$ and measures, at time $t$, the fields generated by the electron at time $t^{\prime }$. The point $O$ is the origin of the reference frame.

Figure 6

LCLS undulator schematic view, showing its various components: undulator sections, focusing $F$, and defocusing $D$, quadrupoles, vacuum pumps, and beam position monitors. From [113].

Figure 7

Radiation electric field $x$, solid line, and $y$, dashed line, components evaluated at a detector on axis at $R_0=10\mathrm{m}$. The undulator has a 3 cm period, 100 periods, and $K=3$.

Figure 8

Ratio of the energy density of the velocity and acceleration fields, for a detector on axis, as a function of the distance $R_0$, in meters, from the undulator entrance, for a 3 cm period, 100 periods, and $K=3$ undulator. The velocity field intensity is about ${10}^8$ times smaller and can be neglected even a few meters from the undulator exit.

Figure 9

Spectrum on axis from a 100 periods helical undulator, normalized to the value at the resonant frequency. The intensity is large only near the resonant frequency in a band $\simeq 1/N_U$.

Figure 10

Intensity at the resonant frequency $\omega =n{\omega }_R$ normalized to that of the first harmonic as a function of $n$. The intensity is evaluated on axis, per unit solid angle and frequency, as given by Eq. (2.67). We assume a planar undulator with 100 periods and an undulator parameter equal to $K=3$.

Figure 11

Relative intensity on axis, near the resonant frequency, as a function of $\mathrm{\Delta }=(\omega -{\omega }_R)/{\omega }_R$, for an undulator with 100 periods, a Gaussian energy distribution with rms energy spreads of 0.001 and 0.003 (dashed line), and 0.01 (dotted line).

Figure 12

Peak brightness for undulators in storage rings and operating FELs. From [63].

Figure 13

The first harmonic coefficient $F_1(K)/2$ in the energy exchange equation (3.23).

Figure 14

Electron trajectories in the undulator, showing a longer path for lower energy electrons with respect to higher energy electrons.

Figure 15

Phase space for the FEL pendulum equation, for a particle with different initial conditions. The stable oscillation point is $\mathrm{\Phi }=\pi$ and $\eta =0$. The two unstable equilibrium points are $\mathrm{\Phi }=0$, $2\pi$, and $\eta =0$. The curve dividing the close and open trajectories is the separatrix. The maximum possible value of the normalized relative energy change is ${\eta }_{\max }=\pm 2\sqrt{2|\widehat{\alpha }|}$.

Figure 16

Evolution along the undulator of the electron distribution in phase space, starting with a monoenergetic beam, ${\eta }_0=0.1$, uniform in phase. The maximum bunching is obtained after one-quarter of the synchrotron period.

Figure 17

Average energy change (solid line), beam energy spread (dashed line), and bunching factor (dotted line), along the undulator. We assume an undulator length corresponding to one-quarter of the synchrotron oscillation period $T$.

Figure 18

The small signal gain function. The function is antisymmetric. The width of the positive peak between 0 and $2\pi$ is about $\mathrm{\Delta }\gamma /\gamma \approx (1/2)N_U$.

Figure 19

The imaginary part of the two complex roots of the dispersion relation.

Figure 20

Dependence of the imaginary part on the beam initial energy spread.

Figure 21

Temporal profiles of a SASE pulse for different values of $\overline{z}$. As the FEL pulse propagates on the electrons, a longitudinal correlation is developed and the width of the temporal spikes grows.

Figure 22

Green’s function for $\overline{z}=10/\sqrt{3}$ (left), $\overline{z}=15/\sqrt{3}$ (center), and $\overline{z}=20/\sqrt{3}$ (right). Notice the difference in the vertical scale.

Figure 23

Normalized radiation field amplitude as a function of interaction time, starting from an initial bunching factor of 0.001.

Figure 24

Longitudinal phase-space trajectories for different values of the interaction time $\overline{z}$ starting from $a_0=0.001$.

Figure 25

Normalized radiation field amplitude at saturation as a function of detuning

Figure 26

Normalized growth rate as a function of detuning for the first, third, and fifth harmonics for a planar undulator with $K=3$.

Figure 27

Maximum normalized growth rate for the first, third, and fifth harmonics as a function of the undulator parameter.

Figure 28

Lowest order dispersion equation solutions for the FEL eigenmodes.

Figure 29

Growth rate of the lowest ${\mathrm{TEM}}_{n,m}$ modes as a function of detuning.

Figure 30

Growth rates for different modes as a function of the Fresnel parameter.

Figure 31

Farfield pattern of a single wave front: (a) after three gain lengths, (b) after seven gain lengths, (c) at saturation, and (d) summation of the far field pattern of all wave fronts within an FEL pulse after three gain lengths.

Figure 32

Overlap between the bunching distribution and radiation wave front for the first higher radial FEL eigenmode. The bunching distribution is distorted by focusing and does not overlap well with the radiation field.

Figure 33

Radiation power of the fundamental mode, the first higher mode with a rigid electron beam and an electron beam in a focusing channel of quadrupoles with alternating polarity channel (dotted, solid, and dashed lines, respectively).

Figure 34

Growth rate of the first higher FEL mode for a rigid and a focussed electron beam (solid and dashed lines, respectively). The oscillation in the dashed line has the same periodicity as the quadrupole spacing of the focusing channel.

Figure 35

Gain-length dependence of the transverse beam focusing beta function for standard LCLS electron beam parameters, using the gain-length fitting formula (6.30). For large values the system is in the 1D limit and stronger focusing gives higher gain by increasing the electron volume density. As beta decreases the longitudinal velocity spread due to the emittance increases, eventually disrupting the FEL gain. The beta function has an optimum when these two effects balance each other.

Figure 36

Tapered undulator potential. Here the synchronous phase is $\pi /6$, and $|\widehat{a}|=0.1$. Electrons execute stable oscillations around the synchronous particle in the areas under the dashed lines.

Figure 37

Tapered undulator phase space. Electrons within the potential well with phases between ${\mathrm{\Phi }}_s$, ${\mathrm{\Phi }}^{*}$ and $\eta <{\eta }_{\max }$ oscillate around the synchronous particle and follow, on average, its energy change.

Figure 38

Comparison of power output for (a) a tapered monochromatic amplifier, (b) a tapered SASE FEL, and (c) a nontapered SASE FEL. Simulations based on the Ginger code ([55]). From [58].

Figure 39

Power and bunching factor as a function of longitudinal distance along the undulator for Gaussian, parabolic, and uniform electron transverse distributions. The wavelength is 1.5 Å and the electron bunch length, FWHM, is 16 fs. The first, exponential regime saturation is at about 40 m. A second saturation is seen around 149 m for the Gaussian case and at a longer distance for the other cases. The stronger bunching factor reduction for the Gaussian beam gives more diffraction and a larger x-ray spot size at the undulator exit. From [52].

Figure 40

Spectra of seeded and SASE FEL at the SCSS Test Facility. The SASE spectrum was scaled in amplitude to fit in the plot. From [112].

Figure 41

The general configuration HGHG for the first proof-of-principle experiment done at the SDL laboratory at Brookhaven National Laboratory. From [47].

Figure 42

Measured spectra of the Fermi FEL using two-step cascaded HGHG and the fresh bunch technique. From [5].

Figure 43

Longitudinal phase-space distribution in EEHG configuration after the (a) first modulator and chicane compressor, the (b) second modulator, and the (c) second chicane. (d) The resulting current profile. From [207].

Figure 44

Schematic layout of a self-seeding configuration.

Figure 45

Sample spectra at LCLS for SASE (red) and self seeded (blue) operation. From [8].

Figure 46

Far field angular distribution of the SASE radiation: (a) measured on the CCD camera and (b) simulated with genesis.

Figure 47

LCLS measurements of power vs undulator length (red points). The blue line is a simulation with the code Genesis. Error bars represent the rms statistical uncertainty in the measured power when averaging 30 beam pulses. The measured power gain length is 3.5 m. From [53].

Figure 48

LCLS measurement of transverse coherence. From [198].

Figure 49

A two-color spectrum of a gain modulated FEL obtained at LCLS. The experimental data and a comparison between theory and experimental data are shown. From [121].

Figure 50

Measured longitudinal phase space of the twin electron bunches with (a) lasing off and (c) lasing on. The current profile and the reconstructed x-ray temporal profile are shown in (b) and (d), respectively. From [122].