Undulator

Undulators ( to undulate = to meander, to move in waves ) and wigglers ( to wiggle = to wiggle) are technical devices for generating synchrotron radiation . They are used in high-energy electron accelerators under the heading of insertion devices .

How it works, construction

Undulator

If an electrically charged particle is deflected by a magnetic field, this physically means that it is accelerated across the direction of flight. A particle moving at almost the speed of light - in practice an electron - emits electromagnetic radiation, synchrotron radiation.

The wiggler and the undulator are rows of dipole magnets that alternate north-south. The special magnet arrangement forces a particle flying through it onto a snake-like, mostly sinusoidal path. The particle emits synchrotron radiation in its average direction of flight. The properties of the radiation depend strongly on the length of the individual magnets, the strength of the magnetic field and the speed, charge and mass of the particle.

There are a number of different designs for wigglers and undulators:

Electromagnetic construction in which either a common iron yoke is enclosed by different normally conducting coils ( magnetic flux density B = 2 Tesla (T)) or superconducting current coils are arranged in a shell ( B > 2 T up to about 12 T) so that they form the alternating dipole field ;
Permanent magnets ( B = 0.8-1.0 T);
Hybrid magnets. Here the magnetic fields of several permanent magnets are guided through a metallic yoke; this achieves a higher flux density than with permanent magnets alone. ( B > 2 T)

Difference between wiggler and undulator

Schematic sketch of an undulator

Wigglers differ from undulators in the type of synchrotron radiation they emit: a wiggler generates a continuous spectrum, an undulator a line spectrum.

This is achieved through different designs. In the wiggler the particles are deflected very strongly in order to achieve high photon energy. For this, stronger magnets are used than in undulators; the magnets in the wiggler are typically arranged as a Halbach array . In addition, the period of the magnetic structure (see schematic sketch) in wigglers is usually greater than in undulators. Due to the high deflection, the radiation lobes generated have a large angle to the axis of the wiggler and therefore do not overlap. As a result, the various radiation lobes cannot interfere with one another and the spectrum of the radiation generated is relatively broad. ${\ displaystyle \ lambda _ {u}}$

In the case of the undulator, the electron path is chosen in such a way that all radiation lobes interfere. This is achieved by a small deflection of the electron path. This leads to lower photon energy, but also to a sharp spectrum and higher brilliance . The opening angle of the generated radiation is also smaller.

The intensity of the emitted radiation in the center of the generated radiation beam (i.e. on the optical axis) is proportional to the number of electrons in the emitting electron bundle (bunch) in both types. In addition, it depends on the number of deflection periods, whereby there is an essential difference between wigglers and undulators: with wigglers the intensity on the optical axis is proportional to the number of deflection periods, with undulators it is proportional to the square of this number.

The transition from the undulator to the wiggler is described by the dimensionless undulator parameter :

{\ displaystyle K = {\ frac {eB \ lambda _ {u}} {2 \ pi mc}} \, {\ text {,}}}

here e is the elementary charge, B the magnetic field strength , the undulator period, m the electron mass and c the speed of light . The undulator period is the distance after which the magnetic field returns to its original value. K describes the strength of the deflection of the electrons. ${\ displaystyle \ lambda _ {u}}$

If it is called a wiggler. The deflection of the electrons is relatively large and the light cones do not constructively overlap. (The intensity is only proportional to the number of undulator periods:) The result is that a relatively broad spectrum is created. ${\ displaystyle K> 1}$ ${\ displaystyle I \ propto N}$
If so, the deflection is small, the light cones generated overlap constructively and one speaks of an undulator. (The intensity is then proportional to the square of the number of Undulatorperioden: ) ${\ displaystyle K \ leq 1}$ ${\ displaystyle I \ propto N ^ {2}}$

The wavelength of the emitted radiation can be calculated using the so-called undulator equation:

{\ displaystyle \ lambda = {\ frac {\ lambda _ {u}} {2 \ gamma ^ {2}}} \ left (1 + {\ frac {K ^ {2}} {2}} + (\ theta \ gamma) ^ {2} \ right),}

here the Lorentz factor , the undulator period, K is the undulator parameter described above and the angle measured from the center of the radiated lobe. ${\ displaystyle \ gamma = {\ frac {1} {\ sqrt {1- \ beta ^ {2}}}}}$ ${\ displaystyle \ lambda _ {u}}$ ${\ displaystyle \ theta}$

Despite its name, the equation applies to both the wiggler and the undulator; K determines which of the two it is.

The preceding expression shows that the short-wave radiation is caused by a Lorentz contraction and the Doppler effect (therefore ) of the undulator periods. ${\ displaystyle {\ frac {\ lambda _ {u}} {2 \ gamma ^ {2}}}}$ ${\ displaystyle \ gamma ^ {2}}$
One recognizes that the emitted radiation has the shortest wavelength in the center and that this increases towards the outside. ${\ displaystyle \ theta}$

In order to generate the same photon energy in an undulator as in a wiggler, the electron bunches must be accelerated to a higher energy. This additional expense is accepted because the emitted radiation has a much higher brilliance and a narrower spectrum. Only undulators are used in modern third and fourth generation synchrotron radiation sources.

The length of an undulator is usually a few meters. If you extend the length to several tens of meters or even several hundred meters, the resulting radiation can interact with the electron packet over the longer distance, and in this way you achieve a special microstructuring in the packet. If this is the case, one speaks of a free-electron laser . The intensity of an FEL is therefore not only proportional to the square of the number of undulator periods ( ), but also proportional to the square of the number of electrons contained in the packet ( ). This increases the brilliance and intensity significantly. ${\ displaystyle \ propto N ^ {2}}$ ${\ displaystyle \ propto n ^ {2}}$

Web links

Uwe Schindler, A superconducting undulator with electrically switchable helicity , 2004 Research Center Karlsruhe, Scientific Reports FZKA 6997 (pdf; 2.0 MB)

Individual evidence

^ H. Motz: Undulators and free-electron lasers . In: Contemporary Physics . tape 20 , no. 5 , September 1, 1979, ISSN 0010-7514 , pp. 547-568 , doi : 10.1080 / 00107517908210921 (English).

[1] H. Motz: Undulators and free-electron lasers . In: Contemporary Physics . tape 20 , no. 5 , September 1, 1979, ISSN 0010-7514 , pp. 547-568 , doi : 10.1080 / 00107517908210921 (English).