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CHAPTER 1
The Physics and Status of X-ray Free-electron Lasers
GIANLUCA GELONI, ZHIRO NG HUANG AND CLAUDIO PELLEGRINI
1.1 Introduction
1.1.1 Early Work on X-ray Lasers and the Development of XFELs
Infrared (IR) and visible lasers were initially developed in the 1960s. Starting from the that time there has been a continued effort to extend the generation of coherent electromagnetic radiation to shorter and shorter wavelengths, with the ultimate goal of reaching the X-ray region. One important reason for these efforts is that an X-ray laser, generating a beam of coherent photons at the angstrom wavelength, would open a new window on the exploration of matter at a length scale corresponding to the atom size (the Bohr radius) and probe in great detail the structure of simple and complex molecules. If, at the same time, the X-ray pulse length could be reduced to a few femtoseconds (the Bohr time, the revolution time of a valence electron around the nucleus), one could also explore the dynamics of atomic and molecular process on their own time scale. The dream of exploring atomic/molecular processes on their natural length and time scale would, for the first time, become a reality and open the exploration of new science for physics, chemistry and biology.
The main impediment in the search for an X-ray laser based on inner level atomic transition are the very short lifetime of excited atom-core quantum energy levels and the larger energy required to excite electrons in the inner core levels. George Chapline and Lowell Wood of Lawrence Livermore National Laboratory, one of the most active research institutes for X-ray laser development, estimated the radiative lifetime of an X-ray laser transition to be about 1 fs times the square of the wavelength in angstroms.
The way out of this difficult situation is offered by the generation of electromagnetic waves from relativistic electron beams and X-ray free-electron lasers (XFELs). A history of their development is found in ref. 4. H ere, we summarize some of the steps that led in recent years to the successful demonstration of XFELs and their operation for novel experimental research in physics, chemistry, biology and materials sciences at the femtosecond/angstrom frontier.
An important step on the way to XFELs was H ans Motz's concept of obtaining nearly monochromatic coherent radiation from relativistic electrons moving through a periodic magnetic array, which he called an undulator magnet, as shown in Figure 1.1. Motz evaluated the wavelength (λ) of the radiation emitted at angle θ respective to the undulator axis by a relativistic electron moving along the axis. For the case of a helical undulator, the wavelength is given by
λ = λU(1 + K2 + γ2θ2)/2γ2, (1.1)
where
K = eBUλU/2πmc2 (1.2)
is called the undulator parameter, λU is the undulator period, typically a few centimeters, BU is the undulator magnetic field on the axis, γ is the electron energy in units of the rest energy mc2. The undulator parameter is the normalized vector potential and is typically of the order one to three. The radiation wavelength can be easily tuned using the quadratic dependence on the electron energy and the undulator parameter.
In 1953 Motz and co-workers observed coherent radiation at Stanford at a millimeter wavelength using a planar undulator with 4 cm period and 3 to 5 MeV electron beam, generated by a linear accelerator, with a bunch length shorter than the radiation wavelength. Raising the beam energy to 100 MeV, they observed incoherent radiation. Using Motz's words: "the mm wave generation might have some practical importance. In this case, it is possible to bunch the electron beam so that groups of electrons radiate coherently. It was shown that the power level may be higher by a factor of the order of a million compared to non coherent radiation ...", can we do the same at a wavelength of about 1 Å? T answer is no, we do not know how to generate an electron beam with all electrons squeezed within λ/10 or separated by λ at X-ray wavelengths. But, in this case, nature is kind to us. Under proper conditions using a free-electron laser (FEL), the electron beam can go through a self-organization process and do just that, as we will see later.
1.1.2 Undulator Radiation Characteristics
There are two main types of undulators used to generate radiation: helical and planar. A detailed description of electron trajectories and of the emitted radiation can be found in ref. 8. A ssume a reference frame with z along the undulator axis and x, y in the transverse directions. Consider a relativistic electron with energy E = mc2γ, longitudinal velocity βz near to one and small transverse velocities βx, βy<< 1. In the case of a helical undulator with period λU, wave number kU = 2π/λU and magnetic field on axis B0, the field near the axis is approximated to the lowest order by Bx = B0 sin(kUz), By = B0 cos(kUz), Bz = 0. The trajectory is a helix of radius a = K/kUγ, where the undulator parameter K, given in eqn (1.2), is typically of the order of one or a few.
The transverse velocities are
βx0(z) = x0/c = -(K/γ)sin(kUz), (1.3)
βy0(z) = y/c = (K/γ)cos(kUz). (1.4)
The longitudinal velocity is
[MATHEMATICAL EXPRESSION OMITTED], (1.5)
having assumed the relativistic factor γ >> 1. Notice that, in this case, the longitudinal velocity is a constant.
To understand the FEL physics, it is important to characterize the radiation emitted by one electron as it traverses an undulator magnet on a trajectory near to the magnet axis. The radiation is peaked at a wavelength
[MATHEMATICAL EXPRESSION OMITTED] (1.6)
and consists of a wave train with a number of wave fronts equal to the number of periods, NU, in the undulator. The length of the radiation pulse is NUλr. For a helical undulator, the radiation is circularly polarized, only the fundamental is present on axis at the wavelength λr [eqn (1.6)], and the harmonics appear off axis. For planar undulators, only one of two transverse components of the magnetic field is present. The electron trajectory is a sinusoid of period λU in the plane perpendicular to the magnetic field. The radiation is plane (linearly) polarized instead of circularly polarized. The longitudinal velocity is not a constant as in the helical case and oscillates at twice the undulator period, leading to the presence of an odd harmonic on the axis. The resonant wavelength is still given by eqn (1.6) if we make the substitution of the undulator parameter [eqn (1.2)] with [MATHEMATICAL EXPRESSION OMITTED].
The undulator parameter is typically of the order of one to three. For a case like the Linac Coherent Light Source (LCLS), the beam energy is about 15 GeV and the undulator is planar, with λU = 3 cm, Krms = 2.8 and NU = 3500. The corresponding wavelength is λ = 0.15 nm. The radiation pulse length and duration for a single electron are NUλ ≈ 0.3 µm and 1 fs.
The spectrum is a sum of harmonics of the fundamental. However, on axis, θ = 0, only the fundamental is present for a helical undulator and is given by
[MATHEMATICAL EXPRESSION OMITTED] (1.7)
where
[MATHEMATICAL EXPRESSION OMITTED] (1.8)
The factor ω2/ωR2 in eqn (1.8) is assumed to be equal to 1, a very good approximation. The full width at half maximum (FWHM) of the radiation line on axis is
Δω/ω = 2.8/πNU. (1.9)
We now estimate the number of photons emitted by one electron in the fundamental line and near the axis, within the line width 1/2NU. Since the frequency depends on the emission angle according to eqn (1.1), to remain within this line width the emission angle must be limited to
[MATHEMATICAL EXPRESSION OMITTED] (1.10)
corresponding to a solid angle of
[MATHEMATICAL EXPRESSION OMITTED] (1.11)
If we consider transversely coherent photons within the phase space area σrσθ = λ/4π, the effective source radius is
[MATHEMATICAL EXPRESSION OMITTED] (1.12)
Multiplying eqn (1.7) by the solid angle and by the line width, we obtain
[MATHEMATICAL EXPRESSION OMITTED] (1.13)
The corresponding number of photons is
[MATHEMATICAL EXPRESSION OMITTED] (1.14)
It is interesting to notice that the number of coherent photons emitted by one electron depends only on the fine structure constant ?and the undulator parameter. For typical values of the undulator parameter, the number is of the order of 0.01, a small value. T his result gives us a simple rule for evaluating the number of coherent photons emitted by one electron in crossing an undulator.
In the case of many electrons, Ne, if there is no correlation between them and they are distributed over a length larger than the wavelength, the total number of photons scales linearly with Ne and is hence about 1% of the number of electrons. If, as in the case of the Motz experiment at millimeter wavelength, all electrons are in a length shorter than the wavelength, the number of photons emitted is about 1% of Ne2. For electron bunches generated in linear accelerators, the number of electrons is typically in the range of 107–109. The difference between the two cases is very large.
1.1.3 Introduction to FELs
The next step was the introduction of the FEL concept in 1971 by John Madey, shown schematically in Figures 1.2 and 1.3. In an FEL an electromagnetic wave, co-propagating with the electron beam, is added to the electron beam-undulator magnet studied by Motz, opening new possibilities. FELs combine the physics and technology of the particle accelerator and lasers to generate electromagnetic radiation with very high brightness. Madey used the Weizsacker–Williams method to calculate the gain due to the induced emission of radiation into a single electromagnetic wave propagating in the same direction of a relativistic electron through a periodic transverse magnetic field. Finite gain is available from the far IR through the visible region, raising the possibility of continuously tunable amplifiers and oscillators at these frequencies with the further possibility of partially coherent radiation sources in the ultraviolet (UV). Considering an extension to the X-ray region or beyond 10 keV, Madey noted in his paper: "The dependence of the gain on the square of the final state wavelength probably precludes the development of steady-state oscillations in the region beyond the ultraviolet ...".
Madey's group demonstrated the feasibility of the FEL concept with two experiments. The first, shown in Figure 1.2, was an FEL amplifier at 10.6 µm. It used a 24 MeV electron beam from a superconducting linear accelerator at Stanford, with a current of 5 to 70 mA. The undulator was of the helical type obtained with a superconducting, right-hand double helix, a period of 3.2 cm and a length of 5.2 m. The single pass gain was as large as 7%. The second, shown in Figure 1.3, was an oscillator operating at a wavelength of 3.4 ?m, a beam energy of 43 MeV, the same helical undulator and an optical cavity that was 12.7 m long.
The two experiments by Madey and co-workers use the two basic configurations of an FEL: an amplifier or oscillator, and an external laser beam that seeds the amplifier. When there is no external seed the electron beam propagating through the undulator generates spontaneous incoherent radiation, as observed by Motz at high electron beam energy and short wavelength. The oscillator can also be seeded or it can start from noise generated by the spontaneous radiation.
The analysis of these experiments was based on Madey's quantum theory of the FEL as stimulated bremsstrahlung, valid only for small changes in the electromagnetic wave intensity, the small signal gain regime. H owever, Madey's small signal gain formula does not depend on P lanck's constant, indicating that the FEL is essentially a classical system. A classical small signal gain theory, developed by William Colson, based on non-quantized Maxwell equations for the field and L orentz equation for the electrons gives the same gain value as Madey's quantum theory.
The small signal gain theory shows that the FEL single pass gain depends on the electron beam six-dimensional phase space density. The scaling with wavelength of the required beam characteristics, as the electron bunch length, transverse phase space density and peak current, together with the lack of good optical cavities in the X-rays spectral region, precludes the possibility of pushing FELs to X-ray wavelengths in the small signal gain regime, as already mentioned.
(Continues…)
Excerpted from "X-Ray Free Electron Lasers"
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