Synchrotron radiation

Synchrotron radiation is electromagnetic radiation emitted by relativistic charged particles undergoing transverse acceleration, typically when a magnetic field bends their paths. For ultrarelativistic particles, emission is concentrated near the instantaneous direction of motion rather than distributed isotropically.

A circulating electron beam therefore loses energy even when the bending magnet itself does no work. Radiofrequency accelerating fields replenish that loss in a storage ring. The extracted radiation is useful as an intense source for spectroscopy and imaging; its angular distribution, polarization and optical spectrum depend on particle energy and magnetic geometry. In synchrotron cooling, those same radiation losses reshape a particle population.

At fixed magnetic field, ultrarelativistic electrons lose energy through synchrotron radiation at a rate proportional to $E^2$. For steady injection $Q(E)\propto E^{-s}$, negligible escape and $s>1$, integrating the energy-space continuity equation gives $N(E)=|\dot{E}{|}^{-1}{\int }_E^{\infty }Q(E^{\prime })d{E}^{\prime }\propto E^{-(s+1)}$. Above the cooling break, the optically thin radiation spectral index therefore steepens from $(s-1)/2$ to $s/2$.

The forward radiation cone sweeps past a distant observer while the source moves closer. The difference between emission time and arrival time shortens the observed pulse to order $\rho /({\gamma }^{3}c)$ and broadens its spectrum.

In a fixed tangent viewing geometry, an electron bunch of longitudinal length $L_b$ passes the same radiating point over approximately $L_b/u$. Different electrons then emit from nearly the same point at different times. The incoherent pulse envelope is the convolution of the single-electron intensity profile with the bunch's passage-time distribution, assuming equal particle trajectories and a linear detector. Its width generally has scale at least $\max \{L_b/u,r/(c{\gamma }^3)\}$. A bunch short compared with the orbit need not be short compared with the intrinsic single-electron pulse.

For an ultrarelativistic circular orbit of curvature radius $\rho$, the conventional critical angular frequency is ${\omega }_c=3{\gamma }^{3}c/(2\rho )$. It sets the broad spectral scale rather than a strict maximum frequency.^[ref]