# FEL operating principle

As the name suggests, 'free' electrons play a key role in a Free-Electron Laser (FEL), in contrast to conventional lasers where the electrons are strongly bound to atoms or molecules that are either in the gas-, liquid- or condensed phase.

Only in this sense the electrons in an FEL are free though, for another key ingredient is a periodically varying magnetic field structure, the undulator, that induces a periodic deflection of the trajectory of the beam of electrons.

Depending on the current density and beam energy, FELs operate in one of three regimes:

- the Raman regime in which the coulomb interaction between the electrons is important, i.e. at low energy and high current density,
- the low-gain Compton regime,
- the high-gain collective regime.

The Raman regime is typical for devices that produce microwaves, whereas FELs aiming at the extreme ultraviolet or x-ray regime, where mirrors with high reflectivity are non-existent, will necessarily have to operate in the high-gain collective regime.

Most FELs, including FELIX, operate in the Compton regime.

A generic layout of Compton FELs

A relativistic beam of electrons, produced by an accelerator, is injected into a resonator, consisting of two high-reflectivity mirrors, around the undulator. The magnetic field of the undulator is perpendicular to the direction of the electron beam and periodically changes polarity a (large) number of times along its length. This will cause a periodic deflection of the electrons while traversing the undulator.

The transverse motion is quite analogous to the oscillatory motion of electrons in a stationary dipole antenna and hence will result in the emission of radiation with a frequency equal to the oscillation frequency. The overall motion of the electrons in the undulator resembles the motion of oscillating electrons in a dipole antenna moving close to the speed of light. This high velocity results in a strong Doppler shift: the frequency of the radiation emitted in the forward direction is typically up-shifted by a factor gamma^{2}, where gamma, the Lorentz factor, is a measure of the electron energy. [For FELIX this up-shift is in the range of 300 to 15,000. Given the 65 mm-period of the undulator, this brings the radiation wavelength in the IR range.] This radiation, referred to as spontaneous emission, is usually very weak though. This is a consequence of the fact that the electrons are typically spread out over an interval that is much larger than the radiation wavelength and will therefore not emit coherently. But on successive roundtrips in the resonator, this weak radiation will be amplified (gain mechanism ) by fresh electrons, until saturation sets in at a power level which is typically 10 million to 100 million times that of the spontaneous emission.

Except for some dynamical effects such as slippage, the time structure of the radiation will mimic that of the electron beam.

Existing Compton FELs cover the wavelength range from 1 mm to 0.2 micron, but emphasis is on the (F)IR where conventional lasers are rare. For a rather complete list of existing and planned FELs, view the The World-Wide Web Virtual Library: Free Electron Laser

*Relativistic** particles*

*Relativistic*

*particles*

A particle is said to be relativistic if its velocity is close to the velocity of light (c=300,000 km per second). This velocity of light is a hard limit: particles with a rest mass, like the electron, can only approach this value, whereas particles without rest mass, like the photon, the 'light particle', always move at this velocity. If a relativistic particle acquires more energy, its velocity will hardly increase but its effective mass will, in accordance with Einsteins famous formula: E=m c ^{2}.

*Doppler shift*

*Doppler shift*

The most common manifestation of the Doppler effect is the change (lowering) of the pitch of the whistle of a train or the siren of a police car as it passes by. The physical principle is sketched in the figure: the distance between the crests of the waves, be it sound waves or electromagnetic waves, is smaller when the source moves in the same direction as the wave than in case the source is at rest, and larger when the source moves in the opposite direction.

As the distance between the crests determines the pitch of the sound waves or the colour of the electromagnetic waves as experienced by an observer, the pitch will be higher and the colour will be shifted to the blue when the source is approaching the observer. For sound waves, the source can move faster than the waves, in which case there will be no sound in the forward direction and a shock wave will be present. This causes the annoying **BANG** one experiences when a plane flies by at a speed in excess of Mach 1. For electromagnetic waves, the energy of the photons depends on the colour: blue photons are more energetic than red photons. So as a result of the Doppler shift, the energy of the radiation emitted by relativistic particles is concentrated in a narrow cone around the forward direction, sometimes referred to as the 'head-light' effect.

*Lorentz factor*

*Lorentz factor*

This factor, usually denoted as gamma, is a very convenient quantity in the theory of relativity. It measures the total energy of a particle (kinetic + rest mass energy) in units of its rest mass energy. As the electron rest mass energy is about 0.5 MeV, gamma is about twice the electron energy given in MeV.

*Resonance condition*

*Resonance condition*

As the radiation moves at the speed of light whereas the electrons can never fully reach it, the radiation will move forward with respect to the electron. The radiation is said to be in resonance with the electron if the radiation advances by an integer number (= harmonic number) of radiation wavelengths relative to the electron, for each oscillation of the electron in the undulator field. This is also the radiation that the electron will emit spontaneously. This condition directly yields the formula given for the spontaneous emission.

*Spontaneous emission*

*Spontaneous emission*

Assuming g , the Lorentz factor, to be large, the wavelength of the spontaneous emission, lambda_{s} is given by:

where lambda_{u} is the period of the magnetic field of the undulator, n the harmonic number and K a measure of the strength of the magnetic field. The factor K, which is usually close to 1, accounts for the reduction of the effective forward velocity of the electrons, and hence the Doppler shift, caused by the induced transverse motion. The intensity of the harmonics (n>1) increases with the K-value. For reasons of symmetry, the even harmonics are generally much weaker than the odd harmonics.

The duration of the radiation pulse is equal to the time it takes the electron to traverse the undulator and the number of cycles in the pulse equals the number of periods, N_{u} , of the undulator. A fundamental theorem states that the product of the duration of a pulse and its spectral bandwidth has a minimum value, from which it follows that the spontaneous emission has a finite bandwidth (see figure for the spectral profile at the fundamental, n=1).

In practice, the spectral width will be somewhat larger as a result of the energy- and angular spread of the electron beam, the finite transverse beam size and imperfections in the magnetic field of the undulator. These constraints get more severe with increasing harmonic number. Therefore, most FELs operate at the fundamental, but sometimes lasing at the third harmonic is used to extend the tuning range to shorter wavelengths.

*Gain mechanism*

*Gain mechanism*

First consider one electron moving through the undulator in the presence of radiation that fulfills the resonance condition (e.g. the spontaneous radiation emitted by similar electrons). The transverse current associated with the transverse motion of the charged electrons, will couple with the transverse electric field of the radiation, causing a net (averaged over a period of the undulator) transfer from the electron's kinetic energy to the field or vice versa and hence a slowing down or acceleration of the electron, which will depend on the relative phases of current and field.

Averaging over a large number of electrons uniformly distributed over all phases, the net energy transfer is zero. Due to the phase-dependent energy transfer and associated velocity modulation, a density modulation on the scale of the wavelength ('micro-bunching') will develop along the undulator however and a net energy transfer becomes possible. Exactly at resonance, the radiation produced by the density modulation is 90-degrees out of phase with the radiation driving the modulation, meaning that an oscillator will not start up under this condition. Slightly off-resonance, both amplitude and phase of the density modulation evolve along the undulator and a net energy transfer will occur: from electrons to the field for radiation that is somewhat red-shifted and vice versa for radiation that is blue-shifted.

The inventor of the FEL, John Madey, showed that the small-signal gain is directly proportional to the derivative of the spontaneous emission spectral profile (see figure for n=1).

As this profile is rather narrow, especially for a large number of undulator periods, and the constant of proportionality is small, a high-quality electron beam (i.e. small energy spread and high current density) is required. In the extreme UV, also the demands on the transverse size and divergence of the beam are very challenging. But in recent years there has been a tremendous progress in electron beam technology which has enabled the development of very successful FEL-based X-ray laser facilities.

Even though the original derivation of Madey's theorem was based on Quantum Mechanics, the FEL-mechanism can easily be explained in classical terms. As such, FELs more closely resemble microwave tubes, such as klystrons and gyrotrons, rather than conventional lasers. This is directly related to the small value of the photon energy as compared to the laser bandwidth, i.e. the electron energy divided by 2 N_{u}. In the IR, this ratio is typically one to a million and even in the extreme UV it is still only one to ten thousand.

*Saturation*

*Saturation*

The resonance condition is directly related to the electron energy and will therefore change when the electron energy changes. Neglecting slippage effects, this means that saturation will occur when the field gets so strong and therefore the energy loss by the electrons along the undulator so large, that at the end of the undulator the radiation that was red-shifted at the undulator entrance, appears blue-shifted to the electrons at the end of the undulator, where hence the energy will start to flow back to the electrons. Obviously, the average phase slip per undulator period at which saturation sets in scales inversely proportional to the number of periods. This also means that the amount of energy that can be extracted from the electron is inversely proportional to the number of periods, in accordance with the width of the gain curve.

*Slippage effects*

*Slippage effects*

As mentioned in the paragraph on the gain mechanism, the electrons entering the undulator first have to interact with the radiation in order for the density modulation, and therefore the gain, to grow. At low radiation intensity, this is a slow process and the electrons will radiate primarily at the end of the undulator, where, according to the resonance condition, the electrons have slipped a distance n.N_{u}.lambda (with n the harmonic number, N_{u} the number of periods and lambda the wavelength) behind the radiation that induced the modulation at that position in the bunch. Especially in the case of short electron bunches (of the order of or less than this slippage distance), the dynamics of the FEL are strongly influenced by the slippage. One manifestation thereof is the so-called 'lethargy' effect: contrary to intuition, the laser will not start-up when the resonator length divided by the speed of light exactly matches the repetition time of the electron bunches ('synchronism condition'). Instead, the resonator length has to be somewhat shorter to compensate for the shift of the centroid of the radiation pulse that is caused by the fact that the trailing edge of the pulse is amplified more than the leading edge. This is illustrated by the figure, showing an example of the measured small-signal gain for FELIX as a function of the cavity length, relative to its synchronous value.

Also shown is the saturated optical power, which doesn't show this lethargy effect, as it clearly peaks close to the synchronous value. This can be understood by realising that the time it takes for the density modulation to develop will reduce as the power grows, and that the shift of the centroid therefore diminishes. For a setting of the cavity length near the peak of the small-signal gain curve, the growth of the power will be fast, but because of reduced overlap of the optical pulse and the electron bunches as the power increases, the power will saturate at a relatively low value. Close to synchronism the opposite holds: the gain will be low because of poor overlap during the early stage of the growth, but will improve as the power increases.

Slippage also modifies the simple picture of saturation given in the preceding paragraph. Because the energy transfer from the electrons to the radiation in the first part of the undulator and re-absorption towards the end affect different parts of the radiation pulse, the front part will experience gain, no matter how intense the pulse, whereas the trailing part will experience loss. So the radiation pulse 'automatically' shortens as the intensity increases, while still extracting net energy from the electrons on each round trip. As a consequence, very intense pulses with a duration appreciably shorter than N_{u}.lambda can be generated. Finally, saturation sets in as a result of the fact that the energy extracted from the electrons is proportional to the magnitude of the electric field of the radiation and therefore to the square root of the pulse energy, whereas the cavity round-trip losses increase linearly with pulse energy.

Even in cases where the electron bunch is much longer than the slippage distance and lethargy is absent, short pulses will still develop naturally. However, in that case the saturated pulse will consist of a number of spikes, separated by a distance close to the slippage distance.