Calculation of coherent radiation

Electromagnetic radiation emitted by electron bunches in accelerators is usually incoherent. However, when the bunch length becomes shorter than the wavelength of the emitted radiation, the waves of the individual electrons interfere constructively. As a result, the intensity of this coherent radiation is enhanced by several orders of magnitude at this wavelength. On one hand, this allows for intense radiation. On the other hand, coherent radiation can act as a ''magnifying glass'', which allows one to study the electron bunch structure. I give a brief derivation of the basic equation for coherent radiation and discuss its properties for a simple Gaussian bunch as well as more complex bunches.