Cherenkov radiation
formula_65 is purely imaginary if formula_68 is real and formula_69. That is, when formula_68 is real, Cherenkov radiation has the condition that formula_71. This is the statement that the speed of the particle must be larger than the phase velocity of electromagnetic fields in the medium at frequency formula_11 in order to have Cherenkov radiation. With this purely imaginary formula_65 condition, formula_74 and the integral can be simplified to: