Generalized emittance invariants

It is noted that the mean-square emittance is useful for analyzing the behavior of beams described by a two-dimensional phase space (or of beams described by a higher-dimensional phase space but for which the various degrees of freedom are uncoupled) because it remains invariant under beam transport through any optical system for which there are no nonlinear forces. Lie algebraic properties of the symplectic group are used to show that, in the general case of a six-dimensional phase space (including possible coupling between all degrees of freedom), the concept of mean-square emittance can be generalized to produce three invariants. These invariants (which can be viewed as eigenemittances) are made out of second-order moments and can be shown to form a complete set. It is concluded that they should be very useful in the analysis of general beam transport. The Lie algebraic methods can be extended to make infinitely many invariants out of cubic and higher-order moments