So far the most brilliant X-rays have been produced in circular electron accelerators called synchrotrons. In early synchrotrons the radiation was produced in the bending magnets of the storage rings. Here the relativistic high-energy electrons (or positrons) are accelerated towards the centre of the ring and emit synchrotron radiation tangentially to their circular orbit. The frequency spectrum is continuous and extends from the microwave region into the optical and X-ray regime, depending on the electron energy.

In modern synchrotron light sources the radiation is produced in undulator magnets. These are periodic arrangements of many short dipole magnets of alternating polarity. The electrons move on a wavelike curve through the undulator, but the overall deflection of the beam is zero. Undulator radiation is far more useful for research than radiation from bending magnets because it is almost monochromatic.

To understand the properties of undulator radiation we need the theory of special relativity. One important consequence of relativity is that the mass of a particle grows when it is accelerated to very high energies. An electron that has traversed a voltage of 500 million volts – the energy is then 500 MeV – has a moving mass which is a thousand times larger than the rest mass. A second important result is relativistic length contraction; a moving bar appears to be shortened. Thirdly, the radiation emitted by a source moving with high speed towards an observer is strongly blue-shifted by the relativistic Doppler effect. This phenomenon is similar to the well-known acoustical Doppler effect; the sound of a motorcycle driving towards you with high speed is shifted to higher frequency, but when the motorcycle moves away from you, the sound is shifted to lower frequency.

Now we put ourselves into a wagon moving through the undulator with the average speed of the electrons, which is very close to the speed of light. As the undulator moves towards us its period appears length-contracted, and the electrons in the wagon oscillate at a high frequency and emit their radiation. The observer in the laboratory sees a source that approaches him with high speed, and the relativistic Doppler effect boosts the frequency more than thousandfold. As a consequence of the length contraction and the Doppler effect, the wavelength of undulator radiation is about a million times shorter than the undulator period. The intensity is concentrated in a narrow spectral range. Different electrons radiate independently, hence the total energy produced by a bunch of N electrons is just N times the radiation energy of one electron.