Correlations and coherence in the microwave instability

We expand upon the solvable model for the microwave instability in an electron storage ring as proposed in Wang (Phys. Rev. E 58 (1998) 984). The Vlasov–Maxwell equations are reduced to an integral eigenvalue problem similar to that which arises in the Karhunen–Loéve expansion theory for stochastic processes (Goodman, Statistical Optics, Wiley, New York, 1985). We derive first- and second-order correlation functions for the electron beam line density for the microwave instability. Using a set of coherent modes the correlation functions are diagonalized. A relationship between the first- and second-order correlation functions is obtained and it is shown to be similar to that of the Hanbury-Brown Twiss experiment (Born and Wolf, Principles of Optics, 6th Edition, Pergamon press, New York, 1980; Loudon, The Quantum Theory of Light, Clarendon press, Oxford, 1983). We obtain an expression for the Wigner distribution in terms of the first-order correlation function and relate this to the power in the electric field. We introduce an entropy-like quantity to characterize the coherence of the instability.