Entropy and emittance of particle and photon beams

The emittance as the available phase space area is defined as the product of the elementary cell area δΩ and exp(S), where S is the normalized entropy of a particle beam. The definition is based on the fact that the factor exp(S) can be interpreted as the number of the occupied cells. For particle beams, a closed expression for the emittance in terms of the phase space distribution function is obtained which is independent of δΩ. To compute the emittance of the radiation beam, it is necessary to find the eigenvalues of the correlation operator. An explicit solution is found for the case of a partially coherent radiation beam which is a stochastic superposition of coherent Gaussian beams with a Gaussian probability distribution. Such a beam is a reasonable model for undulator radiation by beam of electrons. From the requirement that the radiation emittance reproduces the particle beam emittance in the incoherent limit, the elementary cell area δΩ is determined unambiguously to be λ, the radiation wavelength. The emittance in the coherent limit then becomes λ